Average Error: 14.8 → 14.3
Time: 16.6s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{512} - \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)}\right)}{\left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{8}\right) \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{64}}}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{1}{4}}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{\frac{1}{512} - \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)} \cdot \left(\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)} \cdot \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)}\right)}{\left(\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}{8}\right) \cdot \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}{64}}}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{1}{4}}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}
double f(double x) {
        double r4243037 = 1.0;
        double r4243038 = 0.5;
        double r4243039 = x;
        double r4243040 = hypot(r4243037, r4243039);
        double r4243041 = r4243037 / r4243040;
        double r4243042 = r4243037 + r4243041;
        double r4243043 = r4243038 * r4243042;
        double r4243044 = sqrt(r4243043);
        double r4243045 = r4243037 - r4243044;
        return r4243045;
}
↓
double f(double x) {
        double r4243046 = 0.001953125;
        double r4243047 = 0.125;
        double r4243048 = 1.0;
        double r4243049 = x;
        double r4243050 = hypot(r4243048, r4243049);
        double r4243051 = r4243050 * r4243050;
        double r4243052 = r4243050 * r4243051;
        double r4243053 = r4243047 / r4243052;
        double r4243054 = r4243053 * r4243053;
        double r4243055 = r4243053 * r4243054;
        double r4243056 = r4243046 - r4243055;
        double r4243057 = r4243053 + r4243047;
        double r4243058 = r4243057 * r4243053;
        double r4243059 = 0.015625;
        double r4243060 = r4243058 + r4243059;
        double r4243061 = r4243056 / r4243060;
        double r4243062 = 0.5;
        double r4243063 = r4243062 / r4243050;
        double r4243064 = r4243062 + r4243063;
        double r4243065 = r4243063 * r4243064;
        double r4243066 = 0.25;
        double r4243067 = r4243065 + r4243066;
        double r4243068 = r4243061 / r4243067;
        double r4243069 = sqrt(r4243064);
        double r4243070 = r4243069 + r4243048;
        double r4243071 = r4243068 / r4243070;
        return r4243071;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

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

  2. Simplified14.8

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

  4. Applied flip--14.8

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

  5. Simplified14.3

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

  7. Applied flip3--14.3

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

  8. Simplified14.3

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

  9. Simplified14.3

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

  11. Applied flip3--14.3

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

  12. Simplified14.3

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

  13. Simplified14.3

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

  14. Final simplification14.3

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))