text: /*
   * Copyright 2020 Google Inc.
   *
   * Use of this source code is governed by a BSD-style license that can be
   * found in the LICENSE file.
   */
  
  #ifndef GrWangsFormula_DEFINED
  #define GrWangsFormula_DEFINED
  
  #include "include/core/SkPoint.h"
  #include "include/core/SkString.h"
  #include "include/private/SkFloatingPoint.h"
  #include "src/gpu/GrVx.h"
  #include "src/gpu/tessellate/GrVectorXform.h"
  
  // Wang's formula gives the minimum number of evenly spaced (in the parametric sense) line segments
  // that a bezier curve must be chopped into in order to guarantee all lines stay within a distance
  // of "1/precision" pixels from the true curve. Its definition for a bezier curve of degree "n" is
  // as follows:
  //
  //     maxLength = max([length(p[i+2] - 2p[i+1] + p[i]) for (0 <= i <= n-2)])
  //     numParametricSegments = sqrt(maxLength * precision * n*(n - 1)/8)
  //
  // (Goldman, Ron. (2003). 5.6.3 Wang's Formula. "Pyramid Algorithms: A Dynamic Programming Approach
  // to Curves and Surfaces for Geometric Modeling". Morgan Kaufmann Publishers.)
  namespace GrWangsFormula {
  
  // Returns the value by which to multiply length in Wang's formula. (See above.)
  template<int Degree> constexpr float length_term(float precision) {
      return (Degree * (Degree - 1) / 8.f) * precision;
  }
  template<int Degree> constexpr float length_term_pow2(float precision) {
      return ((Degree * Degree) * ((Degree - 1) * (Degree - 1)) / 64.f) * (precision * precision);
  }
  
  SK_ALWAYS_INLINE static float root4(float x) {
      return sqrtf(sqrtf(x));
  }
  
  // Returns nextlog2(sqrt(x)):
  //
  //   log2(sqrt(x)) == log2(x^(1/2)) == log2(x)/2 == log2(x)/log2(4) == log4(x)
  //
  SK_ALWAYS_INLINE static int nextlog4(float x) {
      return (sk_float_nextlog2(x) + 1) >> 1;
  }
  
  // Returns nextlog2(sqrt(sqrt(x))):
  //
  //   log2(sqrt(sqrt(x))) == log2(x^(1/4)) == log2(x)/4 == log2(x)/log2(16) == log16(x)
  //
  SK_ALWAYS_INLINE static int nextlog16(float x) {
      return (sk_float_nextlog2(x) + 3) >> 2;
  }
  
  // Returns Wang's formula, raised to the 4th power, specialized for a quadratic curve.
  SK_ALWAYS_INLINE static float quadratic_pow4(float precision, const SkPoint pts[],
                                               const GrVectorXform& vectorXform = GrVectorXform()) {
      using grvx::float2, skvx::bit_pun;
      float2 p0 = bit_pun<float2>(pts[0]);
      float2 p1 = bit_pun<float2>(pts[1]);
      float2 p2 = bit_pun<float2>(pts[2]);
      float2 v = grvx::fast_madd<2>(-2, p1, p0) + p2;
      v = vectorXform(v);
      float2 vv = v*v;
      return (vv[0] + vv[1]) * length_term_pow2<2>(precision);
  }
  
  // Returns Wang's formula specialized for a quadratic curve.
  SK_ALWAYS_INLINE static float quadratic(float precision, const SkPoint pts[],
                                          const GrVectorXform& vectorXform = GrVectorXform()) {
      return root4(quadratic_pow4(precision, pts, vectorXform));
  }
  
  // Returns the log2 value of Wang's formula specialized for a quadratic curve, rounded up to the
  // next int.
  SK_ALWAYS_INLINE static int quadratic_log2(float precision, const SkPoint pts[],
                                             const GrVectorXform& vectorXform = GrVectorXform()) {
      // nextlog16(x) == ceil(log2(sqrt(sqrt(x))))
      return nextlog16(quadratic_pow4(precision, pts, vectorXform));
  }
  
  // Returns Wang's formula, raised to the 4th power, specialized for a cubic curve.
  SK_ALWAYS_INLINE static float cubic_pow4(float precision, const SkPoint pts[],
                                           const GrVectorXform& vectorXform = GrVectorXform()) {
      using grvx::float4;
      float4 p01 = float4::Load(pts);
      float4 p12 = float4::Load(pts + 1);
      float4 p23 = float4::Load(pts + 2);
      float4 v = grvx::fast_madd<4>(-2, p12, p01) + p23;
      v = vectorXform(v);
      float4 vv = v*v;
      return std::max(vv[0] + vv[1], vv[2] + vv[3]) * length_term_pow2<3>(precision);
  }
  
  // Returns Wang's formula specialized for a cubic curve.
  SK_ALWAYS_INLINE static float cubic(float precision, const SkPoint pts[],
                                      const GrVectorXform& vectorXform = GrVectorXform()) {
      return root4(cubic_pow4(precision, pts, vectorXform));
  }
  
  // Returns the log2 value of Wang's formula specialized for a cubic curve, rounded up to the next
  // int.
  SK_ALWAYS_INLINE static int cubic_log2(float precision, const SkPoint pts[],
                                         const GrVectorXform& vectorXform = GrVectorXform()) {
      // nextlog16(x) == ceil(log2(sqrt(sqrt(x))))
      return nextlog16(cubic_pow4(precision, pts, vectorXform));
  }
  
  // Returns the maximum number of line segments a cubic with the given device-space bounding box size
  // would ever need to be divided into. This is simply a special case of the cubic formula where we
  // maximize its value by placing control points on specific corners of the bounding box.
  SK_ALWAYS_INLINE static float worst_case_cubic(float precision, float devWidth, float devHeight) {
      float k = length_term<3>(precision);
      return sqrtf(2*k * SkVector::Length(devWidth, devHeight));
  }
  
  // Returns the maximum log2 number of line segments a cubic with the given device-space bounding box
  // size would ever need to be divided into.
  SK_ALWAYS_INLINE static int worst_case_cubic_log2(float precision, float devWidth,
                                                    float devHeight) {
      float kk = length_term_pow2<3>(precision);
      // nextlog16(x) == ceil(log2(sqrt(sqrt(x))))
      return nextlog16(4*kk * (devWidth * devWidth + devHeight * devHeight));
  }
  
  // Returns Wang's formula specialized for a conic curve, raised to the second power.
  // Input points should be in projected space, and note tolerance parameter is not "precision".
  //
  // This is not actually due to Wang, but is an analogue from (Theorem 3, corollary 1):
  //   J. Zheng, T. Sederberg. "Estimating Tessellation Parameter Intervals for
  //   Rational Curves and Surfaces." ACM Transactions on Graphics 19(1). 2000.
  SK_ALWAYS_INLINE static float conic_pow2(float tolerance, const SkPoint pts[], float w,
                                           const GrVectorXform& vectorXform = GrVectorXform()) {
      using grvx::dot, grvx::float2, grvx::float4, skvx::bit_pun;
      float2 p0 = vectorXform(bit_pun<float2>(pts[0]));
      float2 p1 = vectorXform(bit_pun<float2>(pts[1]));
      float2 p2 = vectorXform(bit_pun<float2>(pts[2]));
  
      // Compute center of bounding box in projected space
      const float2 C = 0.5f * (skvx::min(skvx::min(p0, p1), p2) + skvx::max(skvx::max(p0, p1), p2));
  
      // Translate by -C. This improves translation-invariance of the formula,
      // see Sec. 3.3 of cited paper
      p0 -= C;
      p1 -= C;
      p2 -= C;
  
      // Compute max length
      const float max_len = sqrtf(std::max(dot(p0, p0), std::max(dot(p1, p1), dot(p2, p2))));
  
      // Compute forward differences
      const float2 dp = grvx::fast_madd<2>(-2 * w, p1, p0) + p2;
      const float dw = fabsf(1 - 2 * w + 1);
  
      // Compute numerator and denominator for parametric step size of linearization
      const float r_minus_eps = std::max(0.f, max_len - tolerance);
      const float min_w = std::min(w, 1.f);
      const float numer = sqrtf(grvx::dot(dp, dp)) + r_minus_eps * dw;
      const float denom = 4 * min_w * tolerance;
  
      // Number of segments = sqrt(numer / denom).
      // This assumes parametric interval of curve being linearized is [t0,t1] = [0, 1].
      // If not, the number of segments is (tmax - tmin) / sqrt(denom / numer).
      return numer / denom;
  }
  
  // Returns the value of Wang's formula specialized for a conic curve.
  SK_ALWAYS_INLINE static float conic(float tolerance, const SkPoint pts[], float w,
                                      const GrVectorXform& vectorXform = GrVectorXform()) {
      return sqrtf(conic_pow2(tolerance, pts, w, vectorXform));
  }
  
  // Returns the log2 value of Wang's formula specialized for a conic curve, rounded up to the next
  // int.
  SK_ALWAYS_INLINE static int conic_log2(float tolerance, const SkPoint pts[], float w,
                                         const GrVectorXform& vectorXform = GrVectorXform()) {
      // nextlog4(x) == ceil(log2(sqrt(x)))
      return nextlog4(conic_pow2(tolerance, pts, w, vectorXform));
  }
  
  // Emits an SKSL function that calculates Wang's formula for the given set of 4 points. The points
  // represent a cubic, or, if 'hasConics' is true and w >= 0, a conic defined by the first 3 points.
  inline static SkString as_sksl(bool hasConics) {
      SkString code;
      code.appendf(R"(
      float length_pow2(float2 v) {
          return dot(v, v);
      }
      float wangs_formula(float parametricPrecision, float2 p0, float2 p1, float2 p2, float2 p3,
                          float w) {
          const float CUBIC_TERM_POW2 = %f;
          float l0 = length_pow2(fma(float2(-2), p1, p2) + p0);
          float l1 = length_pow2(fma(float2(-2), p2, p3) + p1);
          float m = CUBIC_TERM_POW2 * max(l0, l1);)", length_term_pow2<3>(1));
      if (hasConics) {
          // FIXME: Use the better formula from GrWangsFormula::conic().
          code.appendf(R"(
          const float QUAD_TERM_POW2 = %f;
          m = (w >= 0.0) ? QUAD_TERM_POW2 * l0 : m;)", length_term_pow2<2>(1));
      }
      code.append(R"(
          return max(ceil(sqrt(parametricPrecision * sqrt(m))), 1.0);
      })");
      return code;
  }
  
  }  // namespace GrWangsFormula
  
  #endif
  