Add optimized version of logistic function for float. As an example, this is about 50% faster than the existing version on Haswell using AVX.

Eigen/src/Core/functors/UnaryFunctors.h

@@ -850,6 +850,67 @@ struct functor_traits<scalar_logistic_op<T> > {
   };
 };
 
+/** \internal
+  * \brief Template specialization of the logistic function for float.
+  *
+  *  Uses just a 9/10-degree rational interpolant which
+  *  interpolates 1/(1+exp(-x)) - 0.5 up to a couple of ulp in the range
+  *  [-18, 18], outside of which the fl(logistic(x)) = {0|1}. The shifted
+  *  logistic is interpolated because it was easier to make the fit converge.
+  *
+  */
+
+template <>
+struct scalar_logistic_op<float> {
+  EIGEN_EMPTY_STRUCT_CTOR(scalar_logistic_op)
+  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE float operator()(const float& x) const {
+    const float one = 1.0f;
+    return one / (one + numext::exp(-x));
+  }
+
+  template <typename Packet> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
+  Packet packetOp(const Packet& _x) const {
+    // Clamp the inputs to the range [-18, 18] since anything outside
+    // this range is 0.0f or 1.0f in single-precision.
+    const Packet x = pmax(pmin(_x, pset1<Packet>(18.0)), pset1<Packet>(-18.0));
+
+    // The monomial coefficients of the numerator polynomial (odd).
+    const Packet alpha_1 = pset1<Packet>(2.48287947061529e-01);
+    const Packet alpha_3 = pset1<Packet>(8.51377133304701e-03);
+    const Packet alpha_5 = pset1<Packet>(6.08574864600143e-05);
+    const Packet alpha_7 = pset1<Packet>(1.15627324459942e-07);
+    const Packet alpha_9 = pset1<Packet>(4.37031012579801e-11);
+
+    // The monomial coefficients of the denominator polynomial (even).
+    const Packet beta_0 = pset1<Packet>(9.93151921023180e-01);
+    const Packet beta_2 = pset1<Packet>(1.16817656904453e-01);
+    const Packet beta_4 = pset1<Packet>(1.70198817374094e-03);
+    const Packet beta_6 = pset1<Packet>(6.29106785017040e-06);
+    const Packet beta_8 = pset1<Packet>(5.76102136993427e-09);
+    const Packet beta_10 = pset1<Packet>(6.10247389755681e-13);
+
+    // Since the polynomials are odd/even, we need x^2.
+    const Packet x2 = pmul(x, x);
+
+    // Evaluate the numerator polynomial p.
+    Packet p = pmadd(x2, alpha_9, alpha_7);
+    p = pmadd(x2, p, alpha_5);
+    p = pmadd(x2, p, alpha_3);
+    p = pmadd(x2, p, alpha_1);
+    p = pmul(x, p);
+
+    // Evaluate the denominator polynomial p.
+    Packet q = pmadd(x2, beta_10, beta_8);
+    q = pmadd(x2, q, beta_6);
+    q = pmadd(x2, q, beta_4);
+    q = pmadd(x2, q, beta_2);
+    q = pmadd(x2, q, beta_0);
+
+    // Divide the numerator by the denominator and shift it up.
+    return pmax(pmin(padd(pdiv(p, q), pset1<Packet>(0.5)), pset1<Packet>(1.0)),
+                pset1<Packet>(0.0));
+  }
+};
 
 } // end namespace internal