Average Error: 15.3 → 14.9
Time: 2.0m
Precision: 64
\[1.0 - \sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)}\]
↓
\[\frac{\frac{e^{\log \left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot 1.0\right) - \sqrt[3]{\left(\left(\left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right) \cdot \left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)\right) \cdot \left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)\right) \cdot \left(\left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)}\right)}}{\left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot 1.0\right) + \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5} + 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}}{1.0 \cdot \left(\sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5} + 1.0\right) + \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\]
1.0 - \sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)}
↓
\frac{\frac{e^{\log \left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot 1.0\right) - \sqrt[3]{\left(\left(\left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right) \cdot \left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)\right) \cdot \left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)\right) \cdot \left(\left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)}\right)}}{\left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot 1.0\right) + \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5} + 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}}{1.0 \cdot \left(\sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5} + 1.0\right) + \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}
double f(double x) {
        double r5145212 = 1.0;
        double r5145213 = 0.5;
        double r5145214 = x;
        double r5145215 = hypot(r5145212, r5145214);
        double r5145216 = r5145212 / r5145215;
        double r5145217 = r5145212 + r5145216;
        double r5145218 = r5145213 * r5145217;
        double r5145219 = sqrt(r5145218);
        double r5145220 = r5145212 - r5145219;
        return r5145220;
}
↓
double f(double x) {
        double r5145221 = 1.0;
        double r5145222 = r5145221 * r5145221;
        double r5145223 = r5145222 * r5145222;
        double r5145224 = r5145223 * r5145222;
        double r5145225 = r5145224 * r5145221;
        double r5145226 = r5145222 * r5145225;
        double r5145227 = x;
        double r5145228 = hypot(r5145221, r5145227);
        double r5145229 = r5145221 / r5145228;
        double r5145230 = r5145221 + r5145229;
        double r5145231 = 0.5;
        double r5145232 = r5145230 * r5145231;
        double r5145233 = sqrt(r5145232);
        double r5145234 = r5145232 * r5145233;
        double r5145235 = r5145234 * r5145234;
        double r5145236 = r5145235 * r5145235;
        double r5145237 = r5145236 * r5145235;
        double r5145238 = r5145235 * r5145234;
        double r5145239 = r5145237 * r5145238;
        double r5145240 = cbrt(r5145239);
        double r5145241 = r5145226 - r5145240;
        double r5145242 = log(r5145241);
        double r5145243 = exp(r5145242);
        double r5145244 = r5145221 * r5145222;
        double r5145245 = r5145234 + r5145244;
        double r5145246 = r5145234 * r5145245;
        double r5145247 = r5145224 + r5145246;
        double r5145248 = r5145243 / r5145247;
        double r5145249 = r5145233 + r5145221;
        double r5145250 = r5145221 * r5145249;
        double r5145251 = r5145250 + r5145232;
        double r5145252 = r5145248 / r5145251;
        return r5145252;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

    \[1.0 - \sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)}\]
  2. Using strategy rm

  3. Applied flip3--15.6

    \[\leadsto \color{blue}{\frac{{1.0}^{3} - {\left(\sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)}\right)}^{3}}{1.0 \cdot 1.0 + \left(\sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)} + 1.0 \cdot \sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)}\right)}}\]

  4. Simplified15.3

    \[\leadsto \frac{\color{blue}{1.0 \cdot \left(1.0 \cdot 1.0\right) - \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}}}{1.0 \cdot 1.0 + \left(\sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)} + 1.0 \cdot \sqrt{0.5 \cdot \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right)}\right)}\]

  5. Simplified14.9

    \[\leadsto \frac{1.0 \cdot \left(1.0 \cdot 1.0\right) - \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}}{\color{blue}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}}\]
  6. Using strategy rm

  7. Applied flip3--15.3

    \[\leadsto \frac{\color{blue}{\frac{{\left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}^{3} - {\left(\left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}^{3}}{\left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right) + \left(\left(\left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right) \cdot \left(\left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right) + \left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(\left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)\right)}}}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}\]

  8. Simplified15.3

    \[\leadsto \frac{\frac{\color{blue}{\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right)\right) \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}{\left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right) + \left(\left(\left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right) \cdot \left(\left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right) + \left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(\left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right) \cdot \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)\right)}}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}\]

  9. Simplified14.9

    \[\leadsto \frac{\frac{\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right)\right) \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}{\color{blue}{\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) + \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot 1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}\]
  10. Using strategy rm

  11. Applied add-cbrt-cube14.9

    \[\leadsto \frac{\frac{\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right)\right) \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}}{\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) + \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot 1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}\]

  12. Applied add-cbrt-cube14.9

    \[\leadsto \frac{\frac{\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right)\right) \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \color{blue}{\sqrt[3]{\left(\left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)\right) \cdot \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)}} \cdot \sqrt[3]{\left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}{\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) + \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot 1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}\]

  13. Applied cbrt-unprod14.9

    \[\leadsto \frac{\frac{\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right)\right) \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \color{blue}{\sqrt[3]{\left(\left(\left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)\right) \cdot \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)\right) \cdot \left(\left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)}}}{\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) + \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot 1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}\]
  14. Using strategy rm

  15. Applied add-exp-log14.9

    \[\leadsto \frac{\frac{\color{blue}{e^{\log \left(\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right)\right) \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \sqrt[3]{\left(\left(\left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)\right) \cdot \left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)\right) \cdot \left(\left(\left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right) \cdot \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)\right)}\right)}}}{\left(1.0 \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) + \left(\sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot 1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)} \cdot \left(0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)\right)\right)}}{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right) + 1.0 \cdot \left(1.0 + \sqrt{0.5 \cdot \left(\frac{1.0}{\mathsf{hypot}\left(1.0, x\right)} + 1.0\right)}\right)}\]

  16. Final simplification14.9

    \[\leadsto \frac{\frac{e^{\log \left(\left(1.0 \cdot 1.0\right) \cdot \left(\left(\left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot 1.0\right) - \sqrt[3]{\left(\left(\left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right) \cdot \left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)\right) \cdot \left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)\right) \cdot \left(\left(\left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right)\right)}\right)}}{\left(\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(1.0 \cdot 1.0\right) + \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\right) \cdot \left(\left(\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5} + 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}}{1.0 \cdot \left(\sqrt{\left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5} + 1.0\right) + \left(1.0 + \frac{1.0}{\mathsf{hypot}\left(1.0, x\right)}\right) \cdot 0.5}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))