Average Error: 15.6 → 15.2
Time: 17.7s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\sqrt{\frac{1 \cdot \left(1 \cdot 1\right) - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \sqrt{\frac{1 \cdot \left(1 \cdot 1\right) - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\sqrt{\frac{1 \cdot \left(1 \cdot 1\right) - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \sqrt{\frac{1 \cdot \left(1 \cdot 1\right) - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double f(double x) {
        double r8662210 = 1.0;
        double r8662211 = 0.5;
        double r8662212 = x;
        double r8662213 = hypot(r8662210, r8662212);
        double r8662214 = r8662210 / r8662213;
        double r8662215 = r8662210 + r8662214;
        double r8662216 = r8662211 * r8662215;
        double r8662217 = sqrt(r8662216);
        double r8662218 = r8662210 - r8662217;
        return r8662218;
}
↓
double f(double x) {
        double r8662219 = 1.0;
        double r8662220 = r8662219 * r8662219;
        double r8662221 = r8662219 * r8662220;
        double r8662222 = 0.5;
        double r8662223 = x;
        double r8662224 = hypot(r8662219, r8662223);
        double r8662225 = r8662219 / r8662224;
        double r8662226 = r8662219 + r8662225;
        double r8662227 = r8662222 * r8662226;
        double r8662228 = sqrt(r8662227);
        double r8662229 = r8662228 * r8662227;
        double r8662230 = r8662221 - r8662229;
        double r8662231 = r8662219 + r8662228;
        double r8662232 = r8662231 * r8662219;
        double r8662233 = r8662232 + r8662227;
        double r8662234 = r8662230 / r8662233;
        double r8662235 = sqrt(r8662234);
        double r8662236 = r8662235 * r8662235;
        return r8662236;
}

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

    \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
  2. Using strategy rm

  3. Applied flip3--15.9

    \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}\]

  4. Simplified15.6

    \[\leadsto \frac{\color{blue}{\left(1 \cdot 1\right) \cdot 1 - \sqrt{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)} \cdot \left(0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)\right)}}{1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}\]

  5. Simplified15.2

    \[\leadsto \frac{\left(1 \cdot 1\right) \cdot 1 - \sqrt{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)} \cdot \left(0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)\right)}{\color{blue}{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right) + 1 \cdot \left(1 + \sqrt{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)}\right)}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt15.2

    \[\leadsto \color{blue}{\sqrt{\frac{\left(1 \cdot 1\right) \cdot 1 - \sqrt{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)} \cdot \left(0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)\right)}{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right) + 1 \cdot \left(1 + \sqrt{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)}\right)}} \cdot \sqrt{\frac{\left(1 \cdot 1\right) \cdot 1 - \sqrt{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)} \cdot \left(0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)\right)}{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right) + 1 \cdot \left(1 + \sqrt{0.5 \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)}\right)}}}\]

  8. Final simplification15.2

    \[\leadsto \sqrt{\frac{1 \cdot \left(1 \cdot 1\right) - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \sqrt{\frac{1 \cdot \left(1 \cdot 1\right) - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot 1 + 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))