2tan (problem 3.3.2)

Average Error: 36.6 → 16.5

Time: 50.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.157168877465971 \cdot 10^{-289}:\\ \;\;\;\;\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right) + \left(\left(\left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\\ \mathbf{elif}\;\varepsilon \le 1.4726209133031981 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right) + \left(\left(\left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\\ \end{array}\]

C

double f(double x, double eps) {
        double r9651299 = x;
        double r9651300 = eps;
        double r9651301 = r9651299 + r9651300;
        double r9651302 = tan(r9651301);
        double r9651303 = tan(r9651299);
        double r9651304 = r9651302 - r9651303;
        return r9651304;
}

↓

double f(double x, double eps) {
        double r9651305 = eps;
        double r9651306 = -6.157168877465971e-289;
        bool r9651307 = r9651305 <= r9651306;
        double r9651308 = sin(r9651305);
        double r9651309 = r9651308 * r9651308;
        double r9651310 = cos(r9651305);
        double r9651311 = r9651310 * r9651310;
        double r9651312 = r9651309 / r9651311;
        double r9651313 = 1.0;
        double r9651314 = x;
        double r9651315 = sin(r9651314);
        double r9651316 = cos(r9651314);
        double r9651317 = r9651315 / r9651316;
        double r9651318 = r9651317 * r9651317;
        double r9651319 = r9651318 * r9651317;
        double r9651320 = r9651308 / r9651310;
        double r9651321 = r9651320 * r9651312;
        double r9651322 = r9651319 * r9651321;
        double r9651323 = r9651313 - r9651322;
        double r9651324 = r9651312 / r9651323;
        double r9651325 = r9651319 + r9651317;
        double r9651326 = r9651324 * r9651325;
        double r9651327 = r9651317 / r9651323;
        double r9651328 = r9651320 / r9651323;
        double r9651329 = r9651327 + r9651328;
        double r9651330 = r9651321 / r9651323;
        double r9651331 = r9651330 * r9651318;
        double r9651332 = r9651329 + r9651331;
        double r9651333 = r9651332 - r9651317;
        double r9651334 = r9651326 + r9651333;
        double r9651335 = r9651318 * r9651328;
        double r9651336 = r9651334 + r9651335;
        double r9651337 = 1.4726209133031981e-53;
        bool r9651338 = r9651305 <= r9651337;
        double r9651339 = r9651305 + r9651314;
        double r9651340 = r9651305 * r9651339;
        double r9651341 = r9651314 * r9651340;
        double r9651342 = r9651341 + r9651305;
        double r9651343 = r9651338 ? r9651342 : r9651336;
        double r9651344 = r9651307 ? r9651336 : r9651343;
        return r9651344;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.4
Herbie	16.5

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

1. Split input into 2 regimes

2. if eps < -6.157168877465971e-289 or 1.4726209133031981e-53 < eps

   1. Initial program 33.9

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

   3. Applied tan-sum 14.2

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

   5. Applied flip3-- 14.3

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

   6. Applied associate-/r/ 14.3

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

   7. Simplified 14.3

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

   8. Taylor expanded around -inf 14.4

      \[\leadsto \color{blue}{\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{3} \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot \left({\left(\cos \varepsilon\right)}^{3} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}}\]

   9. Simplified 12.8

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

   if -6.157168877465971e-289 < eps < 1.4726209133031981e-53

   1. Initial program 46.3

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

   2. Taylor expanded around 0 30.0

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

   3. Simplified 29.9

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

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.157168877465971 \cdot 10^{-289}:\\ \;\;\;\;\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right) + \left(\left(\left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\\ \mathbf{elif}\;\varepsilon \le 1.4726209133031981 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}\right) + \left(\left(\left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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