Average Error: 36.3 → 9.5
Time: 32.2s
Precision: 64
Internal precision: 2176
\[\tan \left(x + \varepsilon\right) - \tan x\]
⬇
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.2078408487578288 \cdot 10^{-32}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{if}\;\varepsilon \le 7.239888659467981 \cdot 10^{-35}:\\ \;\;\;\;\left({x}^2 \cdot {\varepsilon}^3 + {\varepsilon}^{4} \cdot {x}^3\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x\right)}^3 \cdot {\left(\tan \varepsilon\right)}^3}} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.3
Comparison26.7
Herbie9.5
\[ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes.

  2. if eps < -2.2078408487578288e-32

    1. Initial program 29.3

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

    3. Applied tan-quot 29.1

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

    4. Applied tan-sum 2.5

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

    5. Applied frac-sub 2.6

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

    if -2.2078408487578288e-32 < eps < 7.239888659467981e-35

    1. Initial program 44.9

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

    2. Applied taylor 18.6

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

    3. Taylor expanded around 0 18.6

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

    4. Applied simplify 18.6

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

    if 7.239888659467981e-35 < eps

    1. Initial program 30.2

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

    3. Applied tan-sum 2.4

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

    5. Applied add-cbrt-cube 2.5

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

    6. Applied add-cbrt-cube 2.5

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

    7. Applied cbrt-unprod 2.5

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

Runtime

Time bar (total: 32.2s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066785882 2324371342 4059510649 1466361199 2701357084 1216585281)'
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

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

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