Average Error: 15.6 → 15.1
Time: 3.9s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - e^{2 \cdot \log \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - e^{2 \cdot \log \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
double f(double x) {
        double r190380 = 1.0;
        double r190381 = 0.5;
        double r190382 = x;
        double r190383 = hypot(r190380, r190382);
        double r190384 = r190380 / r190383;
        double r190385 = r190380 + r190384;
        double r190386 = r190381 * r190385;
        double r190387 = sqrt(r190386);
        double r190388 = r190380 - r190387;
        return r190388;
}
↓
double f(double x) {
        double r190389 = 1.0;
        double r190390 = 0.5;
        double r190391 = r190389 - r190390;
        double r190392 = r190389 * r190391;
        double r190393 = r190392 * r190392;
        double r190394 = 2.0;
        double r190395 = x;
        double r190396 = hypot(r190389, r190395);
        double r190397 = r190389 / r190396;
        double r190398 = r190390 * r190397;
        double r190399 = log(r190398);
        double r190400 = r190394 * r190399;
        double r190401 = exp(r190400);
        double r190402 = r190393 - r190401;
        double r190403 = r190389 + r190397;
        double r190404 = r190390 * r190403;
        double r190405 = sqrt(r190404);
        double r190406 = r190389 + r190405;
        double r190407 = r190392 + r190398;
        double r190408 = r190406 * r190407;
        double r190409 = r190402 / r190408;
        return r190409;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.6

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

  3. Applied flip--15.6

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

  4. Simplified15.1

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

  6. Applied flip--15.1

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

  7. Applied associate-/l/15.1

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

  9. Applied add-exp-log15.1

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

  10. Applied add-exp-log15.1

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

  11. Applied div-exp15.1

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

  12. Applied add-exp-log15.1

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

  13. Applied prod-exp15.1

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

  14. Applied add-exp-log15.1

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

  15. Applied add-exp-log15.1

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

  16. Applied div-exp15.1

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

  17. Applied add-exp-log15.1

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

  18. Applied prod-exp15.1

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

  19. Applied prod-exp15.1

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

  20. Simplified15.1

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

  21. Final simplification15.1

    \[\leadsto \frac{\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - e^{2 \cdot \log \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right) + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))