Average Error: 16.1 → 15.8
Time: 15.2s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{\left(\left({\left({1}^{3}\right)}^{3} - {\left({0.5}^{3}\right)}^{3}\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot 1\right)\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(1 + 0.5\right)\right) - \left(1 \cdot 0.5\right) \cdot \left(\left({1}^{6} + \left({\left(1 \cdot 0.5\right)}^{3} + {0.5}^{6}\right)\right) \cdot \left(1 \cdot {1}^{3} - \left(0.5 \cdot \left(1 + 0.5\right)\right) \cdot \left(0.5 \cdot \left(1 + 0.5\right)\right)\right)\right)}{\left({1}^{6} + \left({\left(1 \cdot 0.5\right)}^{3} + {0.5}^{6}\right)\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(1 + 0.5\right)\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + 0.5\right)\right)}}{1 + \sqrt{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot 0.5}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{\left(\left({\left({1}^{3}\right)}^{3} - {\left({0.5}^{3}\right)}^{3}\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot 1\right)\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(1 + 0.5\right)\right) - \left(1 \cdot 0.5\right) \cdot \left(\left({1}^{6} + \left({\left(1 \cdot 0.5\right)}^{3} + {0.5}^{6}\right)\right) \cdot \left(1 \cdot {1}^{3} - \left(0.5 \cdot \left(1 + 0.5\right)\right) \cdot \left(0.5 \cdot \left(1 + 0.5\right)\right)\right)\right)}{\left({1}^{6} + \left({\left(1 \cdot 0.5\right)}^{3} + {0.5}^{6}\right)\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(1 + 0.5\right)\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \left(1 \cdot 1 + 0.5 \cdot \left(1 + 0.5\right)\right)}}{1 + \sqrt{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot 0.5}}
double f(double x) {
        double r166377 = 1.0;
        double r166378 = 0.5;
        double r166379 = x;
        double r166380 = hypot(r166377, r166379);
        double r166381 = r166377 / r166380;
        double r166382 = r166377 + r166381;
        double r166383 = r166378 * r166382;
        double r166384 = sqrt(r166383);
        double r166385 = r166377 - r166384;
        return r166385;
}
↓
double f(double x) {
        double r166386 = 1.0;
        double r166387 = 3.0;
        double r166388 = pow(r166386, r166387);
        double r166389 = pow(r166388, r166387);
        double r166390 = 0.5;
        double r166391 = pow(r166390, r166387);
        double r166392 = pow(r166391, r166387);
        double r166393 = r166389 - r166392;
        double r166394 = x;
        double r166395 = hypot(r166386, r166394);
        double r166396 = r166395 * r166386;
        double r166397 = r166393 * r166396;
        double r166398 = r166386 * r166386;
        double r166399 = r166386 + r166390;
        double r166400 = r166390 * r166399;
        double r166401 = r166398 - r166400;
        double r166402 = r166397 * r166401;
        double r166403 = r166386 * r166390;
        double r166404 = 6.0;
        double r166405 = pow(r166386, r166404);
        double r166406 = pow(r166403, r166387);
        double r166407 = pow(r166390, r166404);
        double r166408 = r166406 + r166407;
        double r166409 = r166405 + r166408;
        double r166410 = r166386 * r166388;
        double r166411 = r166400 * r166400;
        double r166412 = r166410 - r166411;
        double r166413 = r166409 * r166412;
        double r166414 = r166403 * r166413;
        double r166415 = r166402 - r166414;
        double r166416 = r166409 * r166401;
        double r166417 = r166415 / r166416;
        double r166418 = r166398 + r166400;
        double r166419 = r166395 * r166418;
        double r166420 = r166417 / r166419;
        double r166421 = r166386 / r166395;
        double r166422 = r166421 + r166386;
        double r166423 = r166422 * r166390;
        double r166424 = sqrt(r166423);
        double r166425 = r166386 + r166424;
        double r166426 = r166420 / r166425;
        return r166426;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 16.1

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

  3. Applied flip--16.1

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

  4. Simplified15.6

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

  6. Applied flip3--15.6

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

  7. Applied associate-*r/15.6

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

  8. Applied frac-sub15.6

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

  9. Simplified15.6

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

  10. Simplified15.6

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

  12. Applied flip-+15.6

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

  13. Applied associate-*l/15.6

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

  14. Applied flip3--15.6

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

  15. Applied associate-*l/15.8

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

  16. Applied frac-sub15.8

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

  17. Simplified15.8

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

  18. Simplified15.8

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

  19. Final simplification15.8

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))