Average Error: 15.1 → 14.6
Time: 33.0s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{1}{512} - {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{4}\right) \cdot \left(\left(\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{8} + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)\right) + \frac{1}{64}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{1}{512} - {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{4}\right) \cdot \left(\left(\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{8} + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)\right) + \frac{1}{64}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}
double f(double x) {
        double r5661494 = 1.0;
        double r5661495 = 0.5;
        double r5661496 = x;
        double r5661497 = hypot(r5661494, r5661496);
        double r5661498 = r5661494 / r5661497;
        double r5661499 = r5661494 + r5661498;
        double r5661500 = r5661495 * r5661499;
        double r5661501 = sqrt(r5661500);
        double r5661502 = r5661494 - r5661501;
        return r5661502;
}
↓
double f(double x) {
        double r5661503 = 0.001953125;
        double r5661504 = 1.0;
        double r5661505 = x;
        double r5661506 = hypot(r5661504, r5661505);
        double r5661507 = r5661504 / r5661506;
        double r5661508 = 0.125;
        double r5661509 = r5661506 * r5661506;
        double r5661510 = r5661508 / r5661509;
        double r5661511 = r5661507 * r5661510;
        double r5661512 = 3.0;
        double r5661513 = pow(r5661511, r5661512);
        double r5661514 = r5661503 - r5661513;
        double r5661515 = 0.5;
        double r5661516 = r5661515 / r5661506;
        double r5661517 = r5661515 + r5661516;
        double r5661518 = r5661517 * r5661516;
        double r5661519 = 0.25;
        double r5661520 = r5661518 + r5661519;
        double r5661521 = r5661511 * r5661508;
        double r5661522 = r5661511 * r5661511;
        double r5661523 = r5661521 + r5661522;
        double r5661524 = 0.015625;
        double r5661525 = r5661523 + r5661524;
        double r5661526 = r5661520 * r5661525;
        double r5661527 = r5661514 / r5661526;
        double r5661528 = sqrt(r5661517);
        double r5661529 = r5661504 + r5661528;
        double r5661530 = r5661527 / r5661529;
        return r5661530;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.1

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

  2. Simplified15.1

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

  4. Applied flip--15.1

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

  5. Simplified14.6

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

  7. Applied flip3--14.6

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

  8. Simplified14.6

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

  9. Simplified14.6

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

  11. Applied *-un-lft-identity14.6

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

  12. Applied times-frac14.6

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

  14. Applied flip3--14.6

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

  15. Applied associate-/l/14.6

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

  16. Final simplification14.6

    \[\leadsto \frac{\frac{\frac{1}{512} - {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{4}\right) \cdot \left(\left(\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{8} + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)\right) + \frac{1}{64}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

Reproduce

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