2tan (problem 3.3.2)

Average Error: 37.1 → 15.2

Time: 27.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r62228 = x;
        double r62229 = eps;
        double r62230 = r62228 + r62229;
        double r62231 = tan(r62230);
        double r62232 = tan(r62228);
        double r62233 = r62231 - r62232;
        return r62233;
}

↓

double f(double x, double eps) {
        double r62234 = eps;
        double r62235 = -8.360406307762017e-81;
        bool r62236 = r62234 <= r62235;
        double r62237 = x;
        double r62238 = cos(r62237);
        double r62239 = tan(r62234);
        double r62240 = tan(r62237);
        double r62241 = r62239 + r62240;
        double r62242 = sin(r62237);
        double r62243 = -1.0;
        double r62244 = r62240 * r62239;
        double r62245 = r62243 + r62244;
        double r62246 = r62242 * r62245;
        double r62247 = fma(r62238, r62241, r62246);
        double r62248 = 1.0;
        double r62249 = r62248 - r62244;
        double r62250 = r62249 * r62238;
        double r62251 = r62247 / r62250;
        double r62252 = 5.858392615759789e-32;
        bool r62253 = r62234 <= r62252;
        double r62254 = r62237 * r62234;
        double r62255 = r62234 + r62237;
        double r62256 = r62254 * r62255;
        double r62257 = r62256 + r62234;
        double r62258 = r62240 + r62239;
        double r62259 = 6.0;
        double r62260 = pow(r62244, r62259);
        double r62261 = cbrt(r62260);
        double r62262 = r62248 - r62261;
        double r62263 = r62258 / r62262;
        double r62264 = sin(r62234);
        double r62265 = r62242 * r62264;
        double r62266 = cos(r62234);
        double r62267 = r62266 * r62238;
        double r62268 = r62265 / r62267;
        double r62269 = r62248 + r62268;
        double r62270 = -r62240;
        double r62271 = fma(r62263, r62269, r62270);
        double r62272 = r62253 ? r62257 : r62271;
        double r62273 = r62236 ? r62251 : r62272;
        return r62273;
}

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. Using strategy rm

   5. Applied flip-- 46.7

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

   6. Applied associate-/r/ 46.7

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

   7. Applied fma-neg 46.7

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

   8. Taylor expanded around 0 31.1

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

   9. Simplified 30.8

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

   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

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

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)))