Given's Rotation SVD example, simplified

Average Error: 15.2 → 0.2

Time: 18.8s

Precision: 64

Math

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.002150518847943041871040303902873347396962:\\ \;\;\;\;\frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\\ \mathbf{elif}\;x \le 0.001774327303338343181371428158854541834444:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \frac{0.5}{\sqrt{1}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{6} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{\mathsf{fma}\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 0.5 \cdot \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right), {1}^{4}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\\ \end{array}\]

C

double f(double x) {
        double r139358 = 1.0;
        double r139359 = 0.5;
        double r139360 = x;
        double r139361 = hypot(r139358, r139360);
        double r139362 = r139358 / r139361;
        double r139363 = r139358 + r139362;
        double r139364 = r139359 * r139363;
        double r139365 = sqrt(r139364);
        double r139366 = r139358 - r139365;
        return r139366;
}

↓

double f(double x) {
        double r139367 = x;
        double r139368 = -0.002150518847943042;
        bool r139369 = r139367 <= r139368;
        double r139370 = 1.0;
        double r139371 = r139370 * r139370;
        double r139372 = hypot(r139370, r139367);
        double r139373 = r139370 / r139372;
        double r139374 = r139370 + r139373;
        double r139375 = 0.5;
        double r139376 = r139374 * r139375;
        double r139377 = r139371 - r139376;
        double r139378 = exp(r139373);
        double r139379 = log(r139378);
        double r139380 = r139370 + r139379;
        double r139381 = r139375 * r139380;
        double r139382 = sqrt(r139381);
        double r139383 = r139370 + r139382;
        double r139384 = r139377 / r139383;
        double r139385 = 0.0017743273033383432;
        bool r139386 = r139367 <= r139385;
        double r139387 = 0.25;
        double r139388 = 2.0;
        double r139389 = pow(r139367, r139388);
        double r139390 = sqrt(r139370);
        double r139391 = 3.0;
        double r139392 = pow(r139390, r139391);
        double r139393 = r139389 / r139392;
        double r139394 = 4.0;
        double r139395 = pow(r139367, r139394);
        double r139396 = 5.0;
        double r139397 = pow(r139390, r139396);
        double r139398 = r139395 / r139397;
        double r139399 = 0.1875;
        double r139400 = r139375 / r139390;
        double r139401 = fma(r139398, r139399, r139400);
        double r139402 = r139375 - r139401;
        double r139403 = fma(r139387, r139393, r139402);
        double r139404 = r139403 / r139383;
        double r139405 = 6.0;
        double r139406 = pow(r139370, r139405);
        double r139407 = pow(r139376, r139391);
        double r139408 = log1p(r139407);
        double r139409 = expm1(r139408);
        double r139410 = r139406 - r139409;
        double r139411 = fma(r139370, r139370, r139376);
        double r139412 = r139375 * r139411;
        double r139413 = pow(r139370, r139394);
        double r139414 = fma(r139374, r139412, r139413);
        double r139415 = r139410 / r139414;
        double r139416 = r139415 / r139383;
        double r139417 = r139386 ? r139404 : r139416;
        double r139418 = r139369 ? r139384 : r139417;
        return r139418;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.002150518847943042

   1. Initial program 1.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

   4. Simplified 0.1

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
   5. Using strategy rm

   6. Applied add-log-exp 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \left(1 + \color{blue}{\log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)}\right)}}\]

   if -0.002150518847943042 < x < 0.0017743273033383432

   1. Initial program 29.9

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip-- 29.9

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

   4. Simplified 29.9

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
   5. Using strategy rm

   6. Applied add-log-exp 29.9

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \left(1 + \color{blue}{\log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)}\right)}}\]

   7. Taylor expanded around 0 29.9

      \[\leadsto \frac{\color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}} + 0.5\right) - \left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\]

   8. Simplified 0.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \frac{0.5}{\sqrt{1}}\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\]

   if 0.0017743273033383432 < x

   1. Initial program 1.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

   4. Simplified 0.1

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
   5. Using strategy rm

   6. Applied add-log-exp 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \left(1 + \color{blue}{\log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)}\right)}}\]
   7. Using strategy rm

   8. Applied flip3-- 0.1

      \[\leadsto \frac{\color{blue}{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) + \left(1 \cdot 1\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\]

   9. Simplified 0.1

      \[\leadsto \frac{\frac{\color{blue}{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) + \left(1 \cdot 1\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\]

   10. Simplified 0.1

       \[\leadsto \frac{\frac{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 0.5 \cdot \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right), {1}^{4}\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\]
   11. Using strategy rm

   12. Applied expm1-log1p-u 0.1

       \[\leadsto \frac{\frac{{1}^{6} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}}{\mathsf{fma}\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 0.5 \cdot \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right), {1}^{4}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.002150518847943041871040303902873347396962:\\ \;\;\;\;\frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\\ \mathbf{elif}\;x \le 0.001774327303338343181371428158854541834444:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \frac{0.5}{\sqrt{1}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{6} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right)}{\mathsf{fma}\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 0.5 \cdot \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right), {1}^{4}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \log \left(e^{\frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))