Given's Rotation SVD example, simplified

Average Error: 15.3 → 0.2

Time: 47.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.494777107334507744267060314058426229167 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{\left(1 \cdot 1\right)}^{3} \cdot {\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3} \cdot {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{{\left(1 \cdot 1\right)}^{3} \cdot \mathsf{fma}\left(1, {1}^{3}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right) + {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3} \cdot \mathsf{fma}\left(1, {1}^{3}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 6.663327301348302570196957539394588820869 \cdot 10^{-4}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.25}{{\left(\sqrt{1}\right)}^{3}}, {x}^{2}, 0.5 - \mathsf{fma}\left(\frac{0.1875}{{\left(\sqrt{1}\right)}^{5}}, {x}^{4}, \frac{0.5}{\sqrt{1}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\mathsf{fma}\left(1, {1}^{3}, \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \end{array}\]

C

double f(double x) {
        double r305431 = 1.0;
        double r305432 = 0.5;
        double r305433 = x;
        double r305434 = hypot(r305431, r305433);
        double r305435 = r305431 / r305434;
        double r305436 = r305431 + r305435;
        double r305437 = r305432 * r305436;
        double r305438 = sqrt(r305437);
        double r305439 = r305431 - r305438;
        return r305439;
}

↓

double f(double x) {
        double r305440 = x;
        double r305441 = -9.494777107334508e-05;
        bool r305442 = r305440 <= r305441;
        double r305443 = 1.0;
        double r305444 = r305443 * r305443;
        double r305445 = 3.0;
        double r305446 = pow(r305444, r305445);
        double r305447 = r305446 * r305446;
        double r305448 = hypot(r305443, r305440);
        double r305449 = r305443 / r305448;
        double r305450 = r305443 + r305449;
        double r305451 = 0.5;
        double r305452 = r305450 * r305451;
        double r305453 = pow(r305452, r305445);
        double r305454 = r305453 * r305453;
        double r305455 = r305447 - r305454;
        double r305456 = pow(r305443, r305445);
        double r305457 = r305451 * r305450;
        double r305458 = fma(r305443, r305443, r305457);
        double r305459 = r305458 * r305452;
        double r305460 = fma(r305443, r305456, r305459);
        double r305461 = r305446 * r305460;
        double r305462 = r305453 * r305460;
        double r305463 = r305461 + r305462;
        double r305464 = r305455 / r305463;
        double r305465 = sqrt(r305457);
        double r305466 = r305443 + r305465;
        double r305467 = r305464 / r305466;
        double r305468 = 0.0006663327301348303;
        bool r305469 = r305440 <= r305468;
        double r305470 = 0.25;
        double r305471 = sqrt(r305443);
        double r305472 = pow(r305471, r305445);
        double r305473 = r305470 / r305472;
        double r305474 = 2.0;
        double r305475 = pow(r305440, r305474);
        double r305476 = 0.1875;
        double r305477 = 5.0;
        double r305478 = pow(r305471, r305477);
        double r305479 = r305476 / r305478;
        double r305480 = 4.0;
        double r305481 = pow(r305440, r305480);
        double r305482 = r305451 / r305471;
        double r305483 = fma(r305479, r305481, r305482);
        double r305484 = r305451 - r305483;
        double r305485 = fma(r305473, r305475, r305484);
        double r305486 = r305485 / r305466;
        double r305487 = r305446 - r305453;
        double r305488 = log1p(r305459);
        double r305489 = expm1(r305488);
        double r305490 = fma(r305443, r305456, r305489);
        double r305491 = r305487 / r305490;
        double r305492 = r305491 / r305466;
        double r305493 = r305469 ? r305486 : r305492;
        double r305494 = r305442 ? r305467 : r305493;
        return r305494;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -9.494777107334508e-05

   1. Initial program 1.2

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

   3. Applied flip-- 1.2

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

   4. Simplified 0.2

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

   6. Applied flip3-- 0.2

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

   7. Simplified 0.2

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

   9. Applied flip-- 0.2

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

   10. Applied associate-/l/ 0.2

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

   12. Applied distribute-rgt-in 0.2

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

   if -9.494777107334508e-05 < x < 0.0006663327301348303

   1. Initial program 30.0

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

   3. Applied flip-- 30.0

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

   4. Simplified 30.0

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

   5. Taylor expanded around 0 30.0

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

   6. Simplified 0.2

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

   if 0.0006663327301348303 < x

   1. Initial program 1.1

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

   3. Applied flip-- 1.1

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

   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 flip3-- 0.1

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

   7. Simplified 0.1

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

   9. Applied expm1-log1p-u 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.494777107334507744267060314058426229167 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{\left(1 \cdot 1\right)}^{3} \cdot {\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3} \cdot {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{{\left(1 \cdot 1\right)}^{3} \cdot \mathsf{fma}\left(1, {1}^{3}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right) + {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3} \cdot \mathsf{fma}\left(1, {1}^{3}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 6.663327301348302570196957539394588820869 \cdot 10^{-4}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.25}{{\left(\sqrt{1}\right)}^{3}}, {x}^{2}, 0.5 - \mathsf{fma}\left(\frac{0.1875}{{\left(\sqrt{1}\right)}^{5}}, {x}^{4}, \frac{0.5}{\sqrt{1}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\mathsf{fma}\left(1, {1}^{3}, \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))