2tan (problem 3.3.2)

Average Error: 37.0 → 15.4

Time: 33.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.248907778173893262746741592331412477694 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} \cdot \left(\left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos x\right) - \left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \sin x}{\left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.019591388754019802762699104709904265922 \cdot 10^{-64}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan x + \left(\tan \varepsilon \cdot \tan \varepsilon - \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \end{array}\]

C

double f(double x, double eps) {
        double r16767192 = x;
        double r16767193 = eps;
        double r16767194 = r16767192 + r16767193;
        double r16767195 = tan(r16767194);
        double r16767196 = tan(r16767192);
        double r16767197 = r16767195 - r16767196;
        return r16767197;
}

↓

double f(double x, double eps) {
        double r16767198 = eps;
        double r16767199 = -6.248907778173893e-50;
        bool r16767200 = r16767198 <= r16767199;
        double r16767201 = x;
        double r16767202 = tan(r16767201);
        double r16767203 = tan(r16767198);
        double r16767204 = r16767202 + r16767203;
        double r16767205 = 1.0;
        double r16767206 = sin(r16767198);
        double r16767207 = r16767202 * r16767206;
        double r16767208 = cos(r16767198);
        double r16767209 = r16767207 / r16767208;
        double r16767210 = r16767209 * r16767209;
        double r16767211 = r16767205 - r16767210;
        double r16767212 = r16767204 / r16767211;
        double r16767213 = cos(r16767201);
        double r16767214 = r16767211 * r16767213;
        double r16767215 = r16767212 * r16767214;
        double r16767216 = r16767205 - r16767209;
        double r16767217 = sin(r16767201);
        double r16767218 = r16767216 * r16767217;
        double r16767219 = r16767215 - r16767218;
        double r16767220 = r16767216 * r16767213;
        double r16767221 = r16767219 / r16767220;
        double r16767222 = 1.0195913887540198e-64;
        bool r16767223 = r16767198 <= r16767222;
        double r16767224 = r16767201 * r16767198;
        double r16767225 = r16767201 + r16767198;
        double r16767226 = r16767224 * r16767225;
        double r16767227 = r16767198 + r16767226;
        double r16767228 = 3.0;
        double r16767229 = pow(r16767202, r16767228);
        double r16767230 = pow(r16767203, r16767228);
        double r16767231 = r16767229 + r16767230;
        double r16767232 = r16767202 * r16767203;
        double r16767233 = r16767205 - r16767232;
        double r16767234 = r16767202 * r16767202;
        double r16767235 = r16767203 * r16767203;
        double r16767236 = r16767235 - r16767232;
        double r16767237 = r16767234 + r16767236;
        double r16767238 = r16767233 * r16767237;
        double r16767239 = r16767231 / r16767238;
        double r16767240 = r16767239 - r16767202;
        double r16767241 = r16767223 ? r16767227 : r16767240;
        double r16767242 = r16767200 ? r16767221 : r16767241;
        return r16767242;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.1
Herbie	15.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -6.248907778173893e-50

   1. Initial program 29.2

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

   3. Applied tan-sum 3.8

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

   5. Applied tan-quot 3.8

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

   6. Applied associate-*r/ 3.8

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

   8. Applied flip-- 3.9

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

   9. Applied associate-/r/ 3.9

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

   10. Simplified 3.9

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

   12. Applied tan-quot 3.9

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

   13. Applied flip-+ 3.9

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

   14. Applied associate-*r/ 3.9

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

   15. Applied frac-sub 3.9

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

   16. Simplified 3.9

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

   if -6.248907778173893e-50 < eps < 1.0195913887540198e-64

   1. Initial program 46.8

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

   2. Taylor expanded around 0 31.4

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

   3. Simplified 31.1

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

   if 1.0195913887540198e-64 < eps

   1. Initial program 31.2

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

   3. Applied tan-sum 5.1

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

   5. Applied flip3-+ 5.3

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

   6. Applied associate-/l/ 5.3

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

4. Final simplification 15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.248907778173893262746741592331412477694 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} \cdot \left(\left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos x\right) - \left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \sin x}{\left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.019591388754019802762699104709904265922 \cdot 10^{-64}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan x + \left(\tan \varepsilon \cdot \tan \varepsilon - \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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