Error

Target

Derivation

1. Split input into 2 regimes

2. if (- (/ (log (exp (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x)) < -2.1320835721927445e-16 or 3.142285770032561e-17 < (- (/ (log (exp (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x))

   1. Initial program 31.9

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

   3. Applied tan-sum4.2

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

   5. Applied flip3--4.3

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

   6. Applied associate-/r/4.2

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

   7. Applied fma-neg4.2

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

   if -2.1320835721927445e-16 < (- (/ (log (exp (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x)) < 3.142285770032561e-17

   1. Initial program 44.5

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

   2. Taylor expanded around 0 27.9

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

   3. Applied simplify26.7

      \[\leadsto \color{blue}{(\left(\varepsilon \cdot \varepsilon\right) \cdot \left((x \cdot \left(x \cdot \varepsilon\right) + x)_*\right) + \varepsilon)_*}\]
3. Recombined 2 regimes into one program.

4. Applied simplify14.1

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\log \left(e^{\tan \varepsilon + \tan x}\right)}{1 - \tan \varepsilon \cdot \tan x} - \tan x \le -2.1320835721927445 \cdot 10^{-16}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right) + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;\frac{\log \left(e^{\tan \varepsilon + \tan x}\right)}{1 - \tan \varepsilon \cdot \tan x} - \tan x \le 3.142285770032561 \cdot 10^{-17}:\\ \;\;\;\;(\left(\varepsilon \cdot \varepsilon\right) \cdot \left((x \cdot \left(x \cdot \varepsilon\right) + x)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right) + 1\right) + \left(-\tan x\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(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)))