Average Error: 37.8 → 14.6
Time: 1.3m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -4.9186968565525686 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan x \cdot \left(\tan \varepsilon \cdot \sin x + \cos x\right) - \left(\sin x - \tan \varepsilon \cdot \cos x\right)}{\cos x - \tan \varepsilon \cdot \left(\cos x \cdot \tan x\right)}\\ \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 7.1739441105252405 \cdot 10^{-28}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}\right)}^{3}} \cdot \left(1 + \left(\frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon} + \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}\right)\right) - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.8
Target15.3
Herbie14.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -4.9186968565525686e-11

    1. Initial program 36.9

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

    3. Applied tan-sum10.6

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

    5. Applied add-log-exp10.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}} - \tan x\]
    6. Using strategy rm

    7. Applied tan-quot10.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]

    8. Applied frac-sub10.9

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \sin x}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}}\]

    9. Applied simplify10.8

      \[\leadsto \frac{\color{blue}{\tan x \cdot \left(\tan \varepsilon \cdot \sin x + \cos x\right) - \left(\sin x - \tan \varepsilon \cdot \cos x\right)}}{\left(1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)\right) \cdot \cos x}\]

    10. Applied simplify10.7

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

    if -4.9186968565525686e-11 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 7.1739441105252405e-28

    1. Initial program 39.8

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

    2. Taylor expanded around 0 16.3

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

    if 7.1739441105252405e-28 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 36.7

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

    3. Applied tan-sum14.9

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

    5. Applied tan-quot14.9

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

    6. Applied tan-quot14.9

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

    7. Applied frac-times14.9

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

    9. Applied flip3--14.9

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

    10. Applied associate-/r/14.9

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

    11. Applied simplify14.9

      \[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}^{3}}} \cdot \left(1 \cdot 1 + \left(\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} \cdot \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + 1 \cdot \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)\right) - \tan x\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify14.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -4.9186968565525686 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan x \cdot \left(\tan \varepsilon \cdot \sin x + \cos x\right) - \left(\sin x - \tan \varepsilon \cdot \cos x\right)}{\cos x - \tan \varepsilon \cdot \left(\cos x \cdot \tan x\right)}\\ \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 7.1739441105252405 \cdot 10^{-28}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}\right)}^{3}} \cdot \left(1 + \left(\frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon} + \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}\right)\right) - \tan x\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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