2tan (problem 3.3.2)

Average Error: 37.6 → 15.8

Time: 11.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.5307996958657947 \cdot 10^{-32} \lor \neg \left(\varepsilon \le 1.04782928541883701 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}, 1 + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right), -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r141307 = x;
        double r141308 = eps;
        double r141309 = r141307 + r141308;
        double r141310 = tan(r141309);
        double r141311 = tan(r141307);
        double r141312 = r141310 - r141311;
        return r141312;
}

↓

double f(double x, double eps) {
        double r141313 = eps;
        double r141314 = -4.530799695865795e-32;
        bool r141315 = r141313 <= r141314;
        double r141316 = 1.047829285418837e-23;
        bool r141317 = r141313 <= r141316;
        double r141318 = !r141317;
        bool r141319 = r141315 || r141318;
        double r141320 = x;
        double r141321 = tan(r141320);
        double r141322 = tan(r141313);
        double r141323 = r141321 + r141322;
        double r141324 = 1.0;
        double r141325 = fma(r141321, r141322, r141324);
        double r141326 = r141323 * r141325;
        double r141327 = r141321 * r141322;
        double r141328 = r141327 * r141327;
        double r141329 = r141328 * r141328;
        double r141330 = r141324 - r141329;
        double r141331 = r141326 / r141330;
        double r141332 = r141324 + r141328;
        double r141333 = -r141321;
        double r141334 = fma(r141331, r141332, r141333);
        double r141335 = fma(r141333, r141324, r141321);
        double r141336 = r141334 + r141335;
        double r141337 = 2.0;
        double r141338 = pow(r141313, r141337);
        double r141339 = pow(r141320, r141337);
        double r141340 = fma(r141313, r141339, r141313);
        double r141341 = fma(r141338, r141320, r141340);
        double r141342 = r141341 + r141335;
        double r141343 = r141319 ? r141336 : r141342;
        return r141343;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.6
Target	15.4
Herbie	15.8

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

Derivation

1. Split input into 2 regimes

2. if eps < -4.530799695865795e-32 or 1.047829285418837e-23 < eps

   1. Initial program 30.6

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

   3. Applied tan-sum 2.2

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

   5. Applied add-cube-cbrt 2.6

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

   6. Applied flip-- 2.6

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

   7. Applied associate-/r/ 2.6

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

   8. Applied prod-diff 2.6

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

   9. Simplified 2.2

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

   10. Simplified 2.2

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

   12. Applied flip-- 2.3

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

   13. Applied associate-/r/ 2.3

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

   14. Applied fma-neg 2.3

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

   if -4.530799695865795e-32 < eps < 1.047829285418837e-23

   1. Initial program 46.1

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

   3. Applied tan-sum 46.1

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

   5. Applied add-cube-cbrt 46.9

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

   6. Applied flip-- 46.9

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

   7. Applied associate-/r/ 46.9

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

   8. Applied prod-diff 47.0

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

   9. Simplified 47.0

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

   10. Simplified 46.1

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

   12. Applied tan-quot 46.1

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

   13. Applied associate-*r/ 46.1

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

   14. Applied tan-quot 46.1

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

   15. Applied associate-*r/ 46.1

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

   16. Applied frac-times 46.1

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

   17. Simplified 46.1

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

   18. Simplified 46.1

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

   19. Taylor expanded around 0 32.3

       \[\leadsto \color{blue}{\left(x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)\right)} + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   20. Simplified 32.3

       \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)} + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.5307996958657947 \cdot 10^{-32} \lor \neg \left(\varepsilon \le 1.04782928541883701 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}, 1 + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right), -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))