Average Error: 36.8 → 14.2
Time: 1.3m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3943876692052302 \cdot 10^{-41}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{if}\;\varepsilon \le 4.2516327429727565 \cdot 10^{-18}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}}} - \tan x\right) \cdot \left(\tan x + \frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}}}\right)}{\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}}} + \tan x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.8
Target15.1
Herbie14.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.3943876692052302e-41

    1. Initial program 29.7

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

    3. Applied tan-quot29.5

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

    4. Applied tan-sum3.4

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

    5. Applied frac-sub3.4

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

    if -1.3943876692052302e-41 < eps < 4.2516327429727565e-18

    1. Initial program 45.5

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

    2. Taylor expanded around 0 29.0

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

    if 4.2516327429727565e-18 < eps

    1. Initial program 29.8

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

    3. Applied tan-sum1.1

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

    5. Applied clear-num1.2

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

    7. Applied add-cbrt-cube1.2

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

    8. Applied add-cbrt-cube1.3

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

    9. Applied cbrt-unprod1.2

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

    10. Applied simplify1.2

      \[\leadsto \frac{1}{\frac{1 - \sqrt[3]{\color{blue}{{\left(\tan x\right)}^{3} \cdot {\left(\tan \varepsilon\right)}^{3}}}}{\tan x + \tan \varepsilon}} - \tan x\]
    11. Using strategy rm

    12. Applied flip--1.3

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

    13. Applied simplify1.3

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

    14. Applied simplify1.2

      \[\leadsto \frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}}} - \tan x\right) \cdot \left(\tan x + \frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}}}\right)}{\color{blue}{\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan \varepsilon\right)}^{3} \cdot {\left(\tan x\right)}^{3}}} + \tan x}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

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

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