Average Error: 15.6 → 15.1
Time: 2.1m
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 - \frac{\left(\left(\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)}\right) \cdot \sqrt{\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)}}\right) \cdot \left(\left(\left(\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)}\right) \cdot \sqrt{\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)}}\right) \cdot \left(\left(\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)}\right) \cdot \sqrt{\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)}}\right)\right)}{\left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt{\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt{\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt{\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}{\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + \left(1 + \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left(\left(\sqrt{\frac{1}{2} + \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)\right) \cdot \left(1 + \sqrt{\frac{1}{2} + \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)\right)\right) + \left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + \left(1 + \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 - \frac{\left(\left(\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)}\right) \cdot \sqrt{\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)}}\right) \cdot \left(\left(\left(\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)}\right) \cdot \sqrt{\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)}}\right) \cdot \left(\left(\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)}\right) \cdot \sqrt{\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)}}\right)\right)}{\left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt{\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt{\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt{\frac{1}{2} \cdot \left(\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}{\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + \left(1 + \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left(\left(\sqrt{\frac{1}{2} + \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)\right) \cdot \left(1 + \sqrt{\frac{1}{2} + \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)\right)\right) + \left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + \left(1 + \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}
double f(double x) {
        double r10276119 = 1.0;
        double r10276120 = 0.5;
        double r10276121 = x;
        double r10276122 = hypot(r10276119, r10276121);
        double r10276123 = r10276119 / r10276122;
        double r10276124 = r10276119 + r10276123;
        double r10276125 = r10276120 * r10276124;
        double r10276126 = sqrt(r10276125);
        double r10276127 = r10276119 - r10276126;
        return r10276127;
}
↓
double f(double x) {
        double r10276128 = 1.0;
        double r10276129 = 0.125;
        double r10276130 = x;
        double r10276131 = hypot(r10276128, r10276130);
        double r10276132 = r10276131 * r10276131;
        double r10276133 = r10276131 * r10276132;
        double r10276134 = r10276129 / r10276133;
        double r10276135 = r10276129 + r10276134;
        double r10276136 = sqrt(r10276135);
        double r10276137 = r10276135 * r10276136;
        double r10276138 = r10276137 * r10276137;
        double r10276139 = r10276137 * r10276138;
        double r10276140 = 0.5;
        double r10276141 = r10276140 / r10276131;
        double r10276142 = r10276140 - r10276141;
        double r10276143 = r10276140 * r10276142;
        double r10276144 = r10276141 * r10276141;
        double r10276145 = r10276143 + r10276144;
        double r10276146 = sqrt(r10276145);
        double r10276147 = r10276145 * r10276146;
        double r10276148 = r10276147 * r10276147;
        double r10276149 = r10276147 * r10276148;
        double r10276150 = r10276139 / r10276149;
        double r10276151 = r10276128 - r10276150;
        double r10276152 = r10276140 + r10276141;
        double r10276153 = sqrt(r10276152);
        double r10276154 = r10276128 + r10276152;
        double r10276155 = r10276153 + r10276154;
        double r10276156 = r10276153 * r10276152;
        double r10276157 = r10276128 + r10276156;
        double r10276158 = r10276156 * r10276157;
        double r10276159 = r10276155 * r10276158;
        double r10276160 = r10276159 + r10276155;
        double r10276161 = r10276151 / r10276160;
        return r10276161;
}

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

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

  2. Simplified15.6

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

  4. Applied flip3--15.9

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

  5. Simplified15.6

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

  6. Simplified15.1

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

  8. Applied flip3--15.6

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

  9. Applied associate-/l/15.6

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

  10. Simplified15.1

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

  12. Applied flip3-+15.1

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

  13. Applied flip3-+15.1

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

  14. Applied sqrt-div15.1

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

  15. Applied frac-times15.1

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

  16. Applied cube-div15.1

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

  17. Simplified15.1

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

  18. Simplified15.1

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

  19. Final simplification15.1

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))