Given's Rotation SVD example, simplified

Average Error: 15.1 → 0.0

Time: 25.3s

Precision: 64

Math

\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.010775316190533964:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{512} - \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)}{\mathsf{fma}\left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}, \frac{1}{8} + \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}, \frac{1}{64}\right)}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \mathbf{elif}\;x \le 0.012795833219468917:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{5}{32}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-3}{16}\right)\right)}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]

C

double f(double x) {
        double r4601290 = 1.0;
        double r4601291 = 0.5;
        double r4601292 = x;
        double r4601293 = hypot(r4601290, r4601292);
        double r4601294 = r4601290 / r4601293;
        double r4601295 = r4601290 + r4601294;
        double r4601296 = r4601291 * r4601295;
        double r4601297 = sqrt(r4601296);
        double r4601298 = r4601290 - r4601297;
        return r4601298;
}

↓

double f(double x) {
        double r4601299 = x;
        double r4601300 = -0.010775316190533964;
        bool r4601301 = r4601299 <= r4601300;
        double r4601302 = 0.001953125;
        double r4601303 = 0.125;
        double r4601304 = 1.0;
        double r4601305 = hypot(r4601304, r4601299);
        double r4601306 = r4601303 / r4601305;
        double r4601307 = r4601305 * r4601305;
        double r4601308 = r4601306 / r4601307;
        double r4601309 = r4601308 * r4601308;
        double r4601310 = r4601308 * r4601309;
        double r4601311 = r4601302 - r4601310;
        double r4601312 = r4601303 + r4601308;
        double r4601313 = 0.015625;
        double r4601314 = fma(r4601308, r4601312, r4601313);
        double r4601315 = r4601311 / r4601314;
        double r4601316 = 0.5;
        double r4601317 = r4601316 / r4601305;
        double r4601318 = r4601316 + r4601317;
        double r4601319 = 0.25;
        double r4601320 = fma(r4601318, r4601317, r4601319);
        double r4601321 = r4601315 / r4601320;
        double r4601322 = sqrt(r4601318);
        double r4601323 = r4601304 + r4601322;
        double r4601324 = r4601321 / r4601323;
        double r4601325 = 0.012795833219468917;
        bool r4601326 = r4601299 <= r4601325;
        double r4601327 = r4601299 * r4601299;
        double r4601328 = r4601327 * r4601327;
        double r4601329 = r4601328 * r4601327;
        double r4601330 = 0.15625;
        double r4601331 = -0.1875;
        double r4601332 = r4601328 * r4601331;
        double r4601333 = fma(r4601329, r4601330, r4601332);
        double r4601334 = fma(r4601319, r4601327, r4601333);
        double r4601335 = r4601334 / r4601323;
        double r4601336 = expm1(r4601317);
        double r4601337 = log1p(r4601336);
        double r4601338 = r4601316 - r4601337;
        double r4601339 = r4601338 / r4601323;
        double r4601340 = r4601326 ? r4601335 : r4601339;
        double r4601341 = r4601301 ? r4601324 : r4601340;
        return r4601341;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.010775316190533964

   1. Initial program 1.0

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

   2. Simplified 1.0

      \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   3. Using strategy rm

   4. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\]

   5. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   6. Using strategy rm

   7. Applied flip3-- 0.1

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

   8. Simplified 0.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{8} - \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   9. Simplified 0.1

      \[\leadsto \frac{\frac{\frac{1}{8} - \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   10. Using strategy rm

   11. Applied associate-/r* 0.1

       \[\leadsto \frac{\frac{\frac{1}{8} - \color{blue}{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   12. Using strategy rm

   13. Applied flip3-- 0.1

       \[\leadsto \frac{\frac{\color{blue}{\frac{{\frac{1}{8}}^{3} - {\left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\frac{1}{8} \cdot \frac{1}{8} + \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{8} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   14. Simplified 0.1

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{512} - \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{\frac{1}{8} \cdot \frac{1}{8} + \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{8} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   15. Simplified 0.1

       \[\leadsto \frac{\frac{\frac{\frac{1}{512} - \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\color{blue}{\mathsf{fma}\left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}, \frac{1}{8} + \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}, \frac{1}{64}\right)}}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   if -0.010775316190533964 < x < 0.012795833219468917

   1. Initial program 29.4

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

   2. Simplified 29.4

      \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   3. Using strategy rm

   4. Applied flip-- 29.5

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\]

   5. Simplified 29.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   6. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot {x}^{2} + \frac{5}{32} \cdot {x}^{6}\right) - \frac{3}{16} \cdot {x}^{4}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   7. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{5}{32}, \frac{-3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   if 0.012795833219468917 < x

   1. Initial program 1.0

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

   2. Simplified 1.0

      \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   3. Using strategy rm

   4. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\]

   5. Simplified 0.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   6. Using strategy rm

   7. Applied log1p-expm1-u 0.1

      \[\leadsto \frac{\frac{1}{2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.010775316190533964:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{512} - \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)}{\mathsf{fma}\left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}, \frac{1}{8} + \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}, \frac{1}{64}\right)}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \mathbf{elif}\;x \le 0.012795833219468917:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{5}{32}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-3}{16}\right)\right)}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))