Average Error: 15.2 → 14.7
Time: 3.3m
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\sqrt[3]{1 - {\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}} \cdot \left(\sqrt[3]{1 - {\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}} \cdot \sqrt[3]{1 - {\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}\right)}{\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) + 1}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\sqrt[3]{1 - {\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}} \cdot \left(\sqrt[3]{1 - {\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}} \cdot \sqrt[3]{1 - {\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}\right)}{\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)}} + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) + 1}
double f(double x) {
        double r45813225 = 1.0;
        double r45813226 = 0.5;
        double r45813227 = x;
        double r45813228 = hypot(r45813225, r45813227);
        double r45813229 = r45813225 / r45813228;
        double r45813230 = r45813225 + r45813229;
        double r45813231 = r45813226 * r45813230;
        double r45813232 = sqrt(r45813231);
        double r45813233 = r45813225 - r45813232;
        return r45813233;
}
↓
double f(double x) {
        double r45813234 = 1.0;
        double r45813235 = 0.5;
        double r45813236 = x;
        double r45813237 = hypot(r45813234, r45813236);
        double r45813238 = r45813235 / r45813237;
        double r45813239 = r45813235 + r45813238;
        double r45813240 = sqrt(r45813239);
        double r45813241 = 3.0;
        double r45813242 = pow(r45813240, r45813241);
        double r45813243 = r45813234 - r45813242;
        double r45813244 = cbrt(r45813243);
        double r45813245 = r45813244 * r45813244;
        double r45813246 = r45813244 * r45813245;
        double r45813247 = r45813240 * r45813240;
        double r45813248 = r45813247 + r45813240;
        double r45813249 = r45813248 + r45813234;
        double r45813250 = r45813246 / r45813249;
        return r45813250;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.2

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

  2. Simplified15.2

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

  4. Applied flip3--15.4

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

  6. Applied add-cube-cbrt14.7

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

  7. Final simplification14.7

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))