2tan (problem 3.3.2)

Average Error: 37.1 → 15.2

Time: 26.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.360406307762016700753533365740142087902 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan \varepsilon + \tan x, \sin x \cdot \left(-1 + \tan x \cdot \tan \varepsilon\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 5.858392615759789249871478064054441017892 \cdot 10^{-32}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}}, 1 + \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}, -\tan x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r60301 = x;
        double r60302 = eps;
        double r60303 = r60301 + r60302;
        double r60304 = tan(r60303);
        double r60305 = tan(r60301);
        double r60306 = r60304 - r60305;
        return r60306;
}

↓

double f(double x, double eps) {
        double r60307 = eps;
        double r60308 = -8.360406307762017e-81;
        bool r60309 = r60307 <= r60308;
        double r60310 = x;
        double r60311 = cos(r60310);
        double r60312 = tan(r60307);
        double r60313 = tan(r60310);
        double r60314 = r60312 + r60313;
        double r60315 = sin(r60310);
        double r60316 = -1.0;
        double r60317 = r60313 * r60312;
        double r60318 = r60316 + r60317;
        double r60319 = r60315 * r60318;
        double r60320 = fma(r60311, r60314, r60319);
        double r60321 = 1.0;
        double r60322 = r60321 - r60317;
        double r60323 = r60322 * r60311;
        double r60324 = r60320 / r60323;
        double r60325 = 5.858392615759789e-32;
        bool r60326 = r60307 <= r60325;
        double r60327 = r60310 * r60307;
        double r60328 = r60307 + r60310;
        double r60329 = r60327 * r60328;
        double r60330 = r60307 + r60329;
        double r60331 = r60313 + r60312;
        double r60332 = 6.0;
        double r60333 = pow(r60317, r60332);
        double r60334 = cbrt(r60333);
        double r60335 = r60321 - r60334;
        double r60336 = r60331 / r60335;
        double r60337 = sin(r60307);
        double r60338 = r60315 * r60337;
        double r60339 = cos(r60307);
        double r60340 = r60339 * r60311;
        double r60341 = r60338 / r60340;
        double r60342 = r60321 + r60341;
        double r60343 = -r60313;
        double r60344 = fma(r60336, r60342, r60343);
        double r60345 = r60326 ? r60330 : r60344;
        double r60346 = r60309 ? r60324 : r60345;
        return r60346;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
Herbie	15.2

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

Derivation

1. Split input into 3 regimes

2. if eps < -8.360406307762017e-81

   1. Initial program 30.3

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

   3. Applied tan-sum 5.9

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

   5. Applied tan-quot 5.9

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

   6. Applied frac-sub 6.0

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

   7. Simplified 5.9

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

   if -8.360406307762017e-81 < eps < 5.858392615759789e-32

   1. Initial program 46.7

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

   3. Applied tan-sum 46.7

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

   4. Taylor expanded around 0 31.1

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

   5. Simplified 30.8

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

   if 5.858392615759789e-32 < eps

   1. Initial program 30.3

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

   3. Applied tan-sum 2.0

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

   5. Applied flip-- 2.1

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

   6. Applied associate-/r/ 2.1

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

   7. Applied fma-neg 2.1

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

   9. Applied add-cbrt-cube 2.1

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

   10. Applied add-cbrt-cube 2.1

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

   11. Applied cbrt-unprod 2.1

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

   12. Applied add-cbrt-cube 2.1

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

   13. Applied add-cbrt-cube 2.2

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

   14. Applied cbrt-unprod 2.2

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

   15. Applied cbrt-unprod 2.1

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

   16. Simplified 2.1

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \sqrt[3]{\color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{\left(2 \cdot 3\right)}}}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\]

   17. Taylor expanded around inf 2.1

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

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.360406307762016700753533365740142087902 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan \varepsilon + \tan x, \sin x \cdot \left(-1 + \tan x \cdot \tan \varepsilon\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 5.858392615759789249871478064054441017892 \cdot 10^{-32}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}}, 1 + \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}, -\tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))