Given's Rotation SVD example, simplified

Average Error: 14.8 → 0.0

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.012079716481524932:\\ \;\;\;\;\frac{\frac{\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right), \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right)\right)}{\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} - 1} - \frac{1}{8}}{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{elif}\;x \le 0.011674747997097862:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \frac{5}{32} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) - \frac{3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right), \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{-1}{8}\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \end{array}\]

C

double f(double x) {
        double r2561049 = 1.0;
        double r2561050 = 0.5;
        double r2561051 = x;
        double r2561052 = hypot(r2561049, r2561051);
        double r2561053 = r2561049 / r2561052;
        double r2561054 = r2561049 + r2561053;
        double r2561055 = r2561050 * r2561054;
        double r2561056 = sqrt(r2561055);
        double r2561057 = r2561049 - r2561056;
        return r2561057;
}

↓

double f(double x) {
        double r2561058 = x;
        double r2561059 = -0.012079716481524932;
        bool r2561060 = r2561058 <= r2561059;
        double r2561061 = -0.5;
        double r2561062 = 1.0;
        double r2561063 = hypot(r2561062, r2561058);
        double r2561064 = r2561061 / r2561063;
        double r2561065 = r2561064 + r2561062;
        double r2561066 = -1.0;
        double r2561067 = fma(r2561064, r2561064, r2561066);
        double r2561068 = fma(r2561064, r2561067, r2561067);
        double r2561069 = r2561064 - r2561062;
        double r2561070 = r2561068 / r2561069;
        double r2561071 = r2561065 * r2561070;
        double r2561072 = 0.125;
        double r2561073 = r2561071 - r2561072;
        double r2561074 = 0.5;
        double r2561075 = 0.25;
        double r2561076 = fma(r2561074, r2561065, r2561075);
        double r2561077 = fma(r2561065, r2561065, r2561076);
        double r2561078 = r2561073 / r2561077;
        double r2561079 = r2561074 / r2561063;
        double r2561080 = r2561079 + r2561074;
        double r2561081 = sqrt(r2561080);
        double r2561082 = r2561062 + r2561081;
        double r2561083 = r2561078 / r2561082;
        double r2561084 = 0.011674747997097862;
        bool r2561085 = r2561058 <= r2561084;
        double r2561086 = r2561058 * r2561058;
        double r2561087 = 0.15625;
        double r2561088 = r2561086 * r2561086;
        double r2561089 = r2561086 * r2561088;
        double r2561090 = r2561087 * r2561089;
        double r2561091 = fma(r2561075, r2561086, r2561090);
        double r2561092 = 0.1875;
        double r2561093 = r2561092 * r2561088;
        double r2561094 = r2561091 - r2561093;
        double r2561095 = r2561094 / r2561082;
        double r2561096 = r2561065 * r2561065;
        double r2561097 = -0.125;
        double r2561098 = fma(r2561096, r2561065, r2561097);
        double r2561099 = r2561077 / r2561098;
        double r2561100 = r2561062 / r2561099;
        double r2561101 = r2561100 / r2561082;
        double r2561102 = r2561085 ? r2561095 : r2561101;
        double r2561103 = r2561060 ? r2561083 : r2561102;
        return r2561103;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.012079716481524932

   1. Initial program 1.0

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

   2. Simplified 1.0

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.1

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

   7. Applied flip3-- 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

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

   11. Applied flip-+ 0.1

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

   12. Applied associate-*r/ 0.1

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

   13. Simplified 0.1

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

   if -0.012079716481524932 < x < 0.011674747997097862

   1. Initial program 29.1

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

   2. Simplified 29.1

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

   4. Applied flip-- 29.1

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

   5. Simplified 29.1

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

   6. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot {x}^{2} + \frac{5}{32} \cdot {x}^{6}\right) - \frac{3}{16} \cdot {x}^{4}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

   7. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \frac{5}{32} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

   if 0.011674747997097862 < x

   1. Initial program 1.0

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

   2. Simplified 1.0

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.1

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

   7. Applied flip3-- 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

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

   11. Applied clear-num 0.1

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

   12. Simplified 0.1

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, 1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \mathsf{fma}\left(\frac{1}{2}, 1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)\right)}{\mathsf{fma}\left(\left(1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), 1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{-1}{8}\right)}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.012079716481524932:\\ \;\;\;\;\frac{\frac{\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right), \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right)\right)}{\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} - 1} - \frac{1}{8}}{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{elif}\;x \le 0.011674747997097862:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \frac{5}{32} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) - \frac{3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right), \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{-1}{8}\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \end{array}\]

Reproduce

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