2tan (problem 3.3.2)

Average Error: 37.0 → 15.5

Time: 26.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.143684469318108724539290290060606314626 \cdot 10^{-82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \left(-1 + \tan x \cdot \tan \varepsilon\right) \cdot \sin x\right)}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 6.85981628349951691195956386637334831975 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, x \cdot \left(\varepsilon + x\right), \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}{\tan x + \tan \varepsilon}} - \tan x\\ \end{array}\]

C

double f(double x, double eps) {
        double r93320 = x;
        double r93321 = eps;
        double r93322 = r93320 + r93321;
        double r93323 = tan(r93322);
        double r93324 = tan(r93320);
        double r93325 = r93323 - r93324;
        return r93325;
}

↓

double f(double x, double eps) {
        double r93326 = eps;
        double r93327 = -3.143684469318109e-82;
        bool r93328 = r93326 <= r93327;
        double r93329 = x;
        double r93330 = cos(r93329);
        double r93331 = tan(r93329);
        double r93332 = tan(r93326);
        double r93333 = r93331 + r93332;
        double r93334 = -1.0;
        double r93335 = r93331 * r93332;
        double r93336 = r93334 + r93335;
        double r93337 = sin(r93329);
        double r93338 = r93336 * r93337;
        double r93339 = fma(r93330, r93333, r93338);
        double r93340 = -r93332;
        double r93341 = 1.0;
        double r93342 = fma(r93331, r93340, r93341);
        double r93343 = r93342 * r93330;
        double r93344 = r93339 / r93343;
        double r93345 = 6.859816283499517e-40;
        bool r93346 = r93326 <= r93345;
        double r93347 = r93326 + r93329;
        double r93348 = r93329 * r93347;
        double r93349 = fma(r93326, r93348, r93326);
        double r93350 = r93342 / r93333;
        double r93351 = r93341 / r93350;
        double r93352 = r93351 - r93331;
        double r93353 = r93346 ? r93349 : r93352;
        double r93354 = r93328 ? r93344 : r93353;
        return r93354;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.1
Herbie	15.5

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

Derivation

1. Split input into 3 regimes

2. if eps < -3.143684469318109e-82

   1. Initial program 30.7

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

   3. Applied tan-sum 6.6

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

   5. Applied *-un-lft-identity 6.6

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

   6. Applied *-un-lft-identity 6.6

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

   7. Applied times-frac 6.6

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

   8. Simplified 6.6

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

   9. Simplified 6.6

      \[\leadsto 1 \cdot \color{blue}{\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}} - \tan x\]
   10. Using strategy rm

   11. Applied tan-quot 6.6

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

   12. Applied associate-*r/ 6.6

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

   13. Applied frac-sub 6.6

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

   14. Simplified 6.6

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

   if -3.143684469318109e-82 < eps < 6.859816283499517e-40

   1. Initial program 47.2

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

   2. Taylor expanded around 0 31.2

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

   3. Simplified 31.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, x \cdot \left(\varepsilon + x\right), \varepsilon\right)}\]

   if 6.859816283499517e-40 < eps

   1. Initial program 29.6

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

   3. Applied tan-sum 3.2

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

   5. Applied *-un-lft-identity 3.2

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

   6. Applied *-un-lft-identity 3.2

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

   7. Applied times-frac 3.2

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

   8. Simplified 3.2

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

   9. Simplified 3.2

      \[\leadsto 1 \cdot \color{blue}{\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}} - \tan x\]
   10. Using strategy rm

   11. Applied clear-num 3.3

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

4. Final simplification 15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.143684469318108724539290290060606314626 \cdot 10^{-82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \left(-1 + \tan x \cdot \tan \varepsilon\right) \cdot \sin x\right)}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 6.85981628349951691195956386637334831975 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, x \cdot \left(\varepsilon + x\right), \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}{\tan x + \tan \varepsilon}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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