OpenSTA/dcalc/DmpCeff.cc

// OpenSTA, Static Timing Analyzer
// Copyright (c) 2025, Parallax Software, Inc.
// 
// This program is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
// 
// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
// 
// You should have received a copy of the GNU General Public License
// along with this program. If not, see <https://www.gnu.org/licenses/>.
// 
// The origin of this software must not be misrepresented; you must not
// claim that you wrote the original software.
// 
// Altered source versions must be plainly marked as such, and must not be
// misrepresented as being the original software.
// 
// This notice may not be removed or altered from any source distribution.

// "Performance Computation for Precharacterized CMOS Gates with RC Loads",
// Florentin Dartu, Noel Menezes and Lawrence Pileggi, IEEE Transactions
// on Computer-Aided Design of Integrated Circuits and Systems, Vol 15, No 5,
// May 1996, pg 544-553.
//
// The only real change from the paper is that Vl, the measured low
// slew voltage is matched instead of y20 in eqn 12.

#include "DmpCeff.hh"

#include <algorithm> // abs, min
#include <cmath>    // sqrt, log

#include "Report.hh"
#include "Debug.hh"
#include "Units.hh"
#include "TimingArc.hh"
#include "TableModel.hh"
#include "Liberty.hh"
#include "Network.hh"
#include "Sdc.hh"
#include "Parasitics.hh"
#include "DcalcAnalysisPt.hh"
#include "ArcDelayCalc.hh"
#include "FindRoot.hh"
#include "Variables.hh"

namespace sta {

using std::string;
using std::abs;
using std::min;
using std::max;
using std::sqrt;
using std::log;
using std::isnan;
using std::function;

// Tolerance (as a scale of value) for driver parameters (Ceff, delta t, t0).
static const double driver_param_tol = .01;
// Waveform threshold crossing time tolerance (1.0 = 100%).
static const double vth_time_tol = .01;
// A small number used by luDecomp.
static const double tiny_double = 1.0e-20;
// Max iterations for findRoot.
static const int find_root_max_iter = 20;

// Indices of Newton-Raphson parameter vector.
enum DmpParam { t0, dt, ceff };

static const char *dmp_param_index_strings[] = {"t0", "dt", "Ceff"};

// Indices of Newton-Raphson function value vector.
enum DmpFunc { y20, y50, ipi };

static const char *dmp_func_index_strings[] = {"y20", "y50", "Ipi"};

static double
exp2(double x);

class DmpError : public Exception
{
public:
  DmpError(const char *what);
  virtual ~DmpError() {}
  virtual const char *what() const noexcept { return what_; }

private:
  const char *what_;
};

static double
gateModelRd(const LibertyCell *cell,
	    const GateTableModel *gate_model,
	    const RiseFall *rf,
	    double in_slew,
	    double c2,
	    double c1,
	    const Pvt *pvt,
	    bool pocv_enabled);
static void
newtonRaphson(const int max_iter,
	      double x[],
	      const int n,
	      const double x_tol,
	      // eval(state) is called to fill fvec and fjac.
	      function<void ()> eval,
	      // Temporaries supplied by caller.
	      double *fvec,
	      double **fjac,
	      int *index,
	      double *p,
	      double *scale);
static void
luSolve(double **a,
	const int size,
	const int *index,
	double b[]);
static void
luDecomp(double **a,
	 const int size,
	 int *index,
	 double *scale);

////////////////////////////////////////////////////////////////

// Base class for Dartu/Menezes/Pileggi algorithm.
// Derived classes handle different cases of zero values in the Pi model.
class DmpAlg : public StaState
{
public:
  DmpAlg(int nr_order, StaState *sta);
  virtual ~DmpAlg();
  virtual const char *name() = 0;
  // Set driver model and pi model parameters for delay calculation.
  virtual void init(const LibertyLibrary *library,
		    const LibertyCell *drvr_cell,
		    const Pvt *pvt,
		    const GateTableModel *gate_model,
		    const RiseFall *rf,
		    double rd,
		    double in_slew,
		    double c2,
		    double rpi,
		    double c1);
  virtual void gateDelaySlew(// Return values.
                             double &delay,
			     double &slew) = 0;
  virtual void loadDelaySlew(const Pin *load_pin,
			     double elmore,
                             // Return values.
			     ArcDelay &delay,
			     Slew &slew);
  double ceff() { return ceff_; }

  // Given x_ as a vector of input parameters, fill fvec_ with the
  // equations evaluated at x_ and fjac_ with the jabobian evaluated at x_.
  virtual void evalDmpEqns() = 0;
  // Output response to vs(t) ramp driving pi model load.
  void Vo(double t,
          // Return values.
          double &vo,
          double &dol_dt);
  // Load responce to driver waveform.
  void Vl(double t,
          // Return values.
          double &vl,
          double &dvl_dt);

protected:
  // Find driver parameters t0, delta_t, Ceff.
  void findDriverParams(double ceff);
  void gateCapDelaySlew(double cl,
                        // Return values.
			double &delay,
			double &slew);
  void gateDelays(double ceff,
                  // Return values.
		  double &t_vth,
		  double &t_vl,
		  double &slew);
  // Partial derivatives of y(t) (jacobian).
  void dy(double t,
	  double t0,
	  double dt,
	  double cl,
          // Return values.
	  double &dydt0,
	  double &dyddt,
	  double &dydcl);
  double y0dt(double t,
	      double cl);
  double y0dcl(double t,
	       double cl);
  void showX();
  void showFvec();
  void showJacobian();
  void findDriverDelaySlew(// Return values.
                           double &delay,
			   double &slew);
  double findVoCrossing(double vth,
                        double lower_bound,
                        double upper_bound);
  void showVo();
  double findVlCrossing(double vth,
                        double lower_bound,
                        double upper_bound);
  void showVl();
  void fail(const char *reason);

  // Output response to vs(t) ramp driving capacitive load.
  double y(double t,
	   double t0,
	   double dt,
	   double cl);
  // Output response to unit ramp driving capacitive load.
  double y0(double t,
	    double cl);
  // Output response to unit ramp driving pi model load.
  virtual void V0(double t,
                  // Return values.
                  double &vo,
                  double &dvo_dt) = 0;
  // Upper bound on time that vo crosses vh.
  virtual double voCrossingUpperBound() = 0;
  // Load responce to driver unit ramp.
  virtual void Vl0(double t,
                   // Return values.
                   double &vl,
                   double &dvl_dt) = 0;
  // Upper bound on time that vl crosses vh.
  double vlCrossingUpperBound();

  // Inputs to the delay calculator.
  const LibertyCell *drvr_cell_;
  const LibertyLibrary *drvr_library_;
  const Pvt *pvt_;
  const GateTableModel *gate_model_;
  double in_slew_;
  double c2_;
  double rpi_;
  double c1_;

  double rd_;
  // Logic threshold (percentage of supply voltage).
  double vth_;
  // Slew lower limit (percentage of supply voltage).
  double vl_;
  // Slew upper limit (percentage of supply voltage).
  double vh_;
  // Table slews are scaled by slew_derate to get
  // measured slews from vl to vh.
  double slew_derate_;

  // Driver parameters calculated by this algorithm.
  double t0_;
  double dt_;
  double ceff_;

  // Driver parameter Newton-Raphson state.
  int nr_order_;

  static constexpr int max_nr_order_ = 3;

  double x_[max_nr_order_];
  double fvec_[max_nr_order_];
  double fjac_storage_[max_nr_order_ * max_nr_order_];
  double *fjac_[max_nr_order_];
  double scale_[max_nr_order_];
  double p_[max_nr_order_ ];
  int index_[max_nr_order_];

  // Driver slew used to check load delay.
  double drvr_slew_;
  double vo_delay_;
  // True if the driver parameters are valid for finding the load delays.
  bool driver_valid_;
  // Load rspf elmore delay.
  double elmore_;
  double p3_;
};

DmpAlg::DmpAlg(int nr_order,
	       StaState *sta):
  StaState(sta),
  c2_(0.0),
  rpi_(0.0),
  c1_(0.0),
  nr_order_(nr_order)
{
  for (int i = 0; i < nr_order_; i++)
    // Only use the upper left block of the matrix
    fjac_[i] = fjac_storage_ + i * max_nr_order_;
}

DmpAlg::~DmpAlg() = default;

void
DmpAlg::init(const LibertyLibrary *drvr_library,
	     const LibertyCell *drvr_cell,
	     const Pvt *pvt,
	     const GateTableModel *gate_model,
	     const RiseFall *rf,
	     double rd,
	     double in_slew,
	     // Pi model.
	     double c2,
	     double rpi,
	     double c1)
{
  drvr_library_ = drvr_library;
  drvr_cell_ = drvr_cell;
  pvt_ = pvt;
  gate_model_ = gate_model;
  rd_ = rd;
  in_slew_ = in_slew;
  c2_ = c2;
  rpi_ = rpi;
  c1_ = c1;
  driver_valid_ = false;
  vth_ = drvr_library->outputThreshold(rf);
  vl_ = drvr_library->slewLowerThreshold(rf);
  vh_ = drvr_library->slewUpperThreshold(rf);
  slew_derate_ = drvr_library->slewDerateFromLibrary();
}

// Find Ceff, delta_t and t0 for the driver.
void
DmpAlg::findDriverParams(double ceff)
{
  if (nr_order_ == 3)
    x_[DmpParam::ceff] = ceff;
  double t_vth, t_vl, slew;
  gateDelays(ceff, t_vth, t_vl, slew);
  // Scale slew to 0-100%
  double dt = slew / (vh_ - vl_);
  double t0 = t_vth + log(1.0 - vth_) * rd_ * ceff - vth_ * dt;
  x_[DmpParam::dt] = dt;
  x_[DmpParam::t0] = t0;
  newtonRaphson(100, x_, nr_order_, driver_param_tol,
                [this] () { evalDmpEqns(); },
		fvec_, fjac_, index_, p_, scale_);
  t0_ = x_[DmpParam::t0];
  dt_ = x_[DmpParam::dt];
  debugPrint(debug_, "dmp_ceff", 3, "    t0 = %s dt = %s ceff = %s",
             units_->timeUnit()->asString(t0_),
             units_->timeUnit()->asString(dt_),
             units_->capacitanceUnit()->asString(x_[DmpParam::ceff]));
  if (debug_->check("dmp_ceff", 4))
    showVo();
}

void
DmpAlg::gateCapDelaySlew(double ceff,
			 // Return values.
                         double &delay,
			 double &slew)
{
  ArcDelay model_delay;
  Slew model_slew;
  gate_model_->gateDelay(pvt_, in_slew_, ceff,
                         variables_->pocvEnabled(),
                         model_delay, model_slew);
  delay = delayAsFloat(model_delay);
  slew = delayAsFloat(model_slew);
}

void
DmpAlg::gateDelays(double ceff,
		   // Return values.
                   double &t_vth,
		   double &t_vl,
		   double &slew)
{
  double table_slew;
  gateCapDelaySlew(ceff, t_vth, table_slew);
  // Convert reported/table slew to measured slew.
  slew = table_slew * slew_derate_;
  t_vl = t_vth - slew * (vth_ - vl_) / (vh_ - vl_);
}

double
DmpAlg::y(double t,
	  double t0,
	  double dt,
	  double cl)
{
  double t1 = t - t0;
  if (t1 <= 0.0)
    return 0.0;
  else if (t1 <= dt)
    return y0(t1, cl) / dt;
  else
    return (y0(t1, cl) - y0(t1 - dt, cl)) / dt;
}

double
DmpAlg::y0(double t,
	   double cl)
{
  return t - rd_ * cl * (1.0 - exp2(-t / (rd_ * cl)));
}

void
DmpAlg::dy(double t,
	   double t0,
	   double dt,
	   double cl,
	   // Return values.
	   double &dydt0,
	   double &dyddt,
	   double &dydcl)
{
  double t1 = t - t0;
  if (t1 <= 0.0)
    dydt0 = dyddt = dydcl = 0.0;
  else if (t1 <= dt) {
    dydt0 = -y0dt(t1, cl) / dt;
    dyddt = -y0(t1, cl) / (dt * dt);
    dydcl = y0dcl(t1, cl) / dt;
  }
  else {
    dydt0 = -(y0dt(t1, cl) - y0dt(t1 - dt, cl)) / dt;
    dyddt = -(y0(t1, cl) + y0(t1 - dt, cl)) / (dt * dt)
      + y0dt(t1 - dt, cl) / dt;
    dydcl = (y0dcl(t1, cl) - y0dcl(t1 - dt, cl)) / dt;
  }
}

double
DmpAlg::y0dt(double t,
	     double cl)
{
  return 1.0 - exp2(-t / (rd_ * cl));
}

double
DmpAlg::y0dcl(double t,
	      double cl)
{
  return rd_ * ((1.0 + t / (rd_ * cl)) * exp2(-t / (rd_ * cl)) - 1);
}

void
DmpAlg::showX()
{
  for (int i = 0; i < nr_order_; i++)
    report_->reportLine("%4s %12.3e", dmp_param_index_strings[i], x_[i]);
}

void
DmpAlg::showFvec()
{
  for (int i = 0; i < nr_order_; i++)
    report_->reportLine("%4s %12.3e", dmp_func_index_strings[i], fvec_[i]);
}

void
DmpAlg::showJacobian()
{
  string line = "    ";
  for (int j = 0; j < nr_order_; j++)
    line += stdstrPrint("%12s", dmp_param_index_strings[j]);
  report_->reportLineString(line);
  line.clear();
  for (int i = 0; i < nr_order_; i++) {
    line += stdstrPrint("%4s ", dmp_func_index_strings[i]);
    for (int j = 0; j < nr_order_; j++)
      line += stdstrPrint("%12.3e ", fjac_[i][j]);
    report_->reportLineString(line);
  }
}

void
DmpAlg::findDriverDelaySlew(// Return values.
                            double &delay,
			    double &slew)
{
  double t_upper = voCrossingUpperBound();
  delay = findVoCrossing(vth_, t0_, t_upper);
  double tl = findVoCrossing(vl_, t0_, delay);
  double th = findVoCrossing(vh_, delay, t_upper);
  // Convert measured slew to table slew.
  slew = (th - tl) / slew_derate_;
}

// Find t such that vo(t)=v.
double
DmpAlg::findVoCrossing(double vth,
                       double t_lower,
                       double t_upper)
{
  FindRootFunc vo_func = [&] (double t,
                              double &y,
                              double &dy) {
    double vo, vo_dt;
    Vo(t, vo, vo_dt);
    y = vo - vth;
    dy = vo_dt;
  };
  bool fail;
  double t_vth = findRoot(vo_func, t_lower, t_upper, vth_time_tol,
                          find_root_max_iter, fail);
  if (fail)
    throw DmpError("find Vo crossing failed");
  return t_vth;
}

void
DmpAlg::Vo(double t,
           // Return values.
           double &vo,
           double &dvo_dt)
{
  double t1 = t - t0_;
  if (t1 <= 0.0) {
    vo = 0.0;
    dvo_dt = 0.0;
  }
  else if (t1 <= dt_) {
    double v0, dv0_dt;
    V0(t1, v0, dv0_dt);

    vo = v0 / dt_;
    dvo_dt = dv0_dt / dt_;    
  }
  else {
    double v0, dv0_dt;
    V0(t1, v0, dv0_dt);

    double v0_dt, dv0_dt_dt;
    V0(t1 - dt_, v0_dt, dv0_dt_dt);

    vo = (v0 - v0_dt) / dt_;
    dvo_dt = (dv0_dt - dv0_dt_dt) / dt_;
  }
}

void
DmpAlg::showVo()
{
  report_->reportLine("  t    vo(t)");
  double ub = voCrossingUpperBound();
  for (double t = t0_; t < t0_ + ub; t += dt_ / 10.0) {
    double vo, dvo_dt;
    Vo(t, vo, dvo_dt);
    report_->reportLine(" %g %g", t, vo);
  }
}

void
DmpAlg::loadDelaySlew(const Pin *,
		      double elmore,
		      ArcDelay &delay,
		      Slew &slew)
{
  if (!driver_valid_
      || elmore == 0.0
      // Elmore delay is small compared to driver slew.
      || elmore < drvr_slew_ * 1e-3) {
    delay = elmore;
    slew = drvr_slew_;
  }
  else {
    // Use the driver thresholds and rely on thresholdAdjust to
    // convert the delay and slew to the load's thresholds.
    try {
      if (debug_->check("dmp_ceff", 4))
	showVl();
      elmore_ = elmore;
      p3_ = 1.0 / elmore;
      double t_lower = t0_;
      double t_upper = vlCrossingUpperBound();
      double load_delay = findVlCrossing(vth_, t_lower, t_upper);
      double tl = findVlCrossing(vl_, t_lower, load_delay);
      double th = findVlCrossing(vh_, load_delay, t_upper);
      // Measure delay from Vo, the load dependent source excitation.
      double delay1 = load_delay - vo_delay_;
      // Convert measured slew to reported/table slew.
      double slew1 = (th - tl) / slew_derate_;
      if (delay1 < 0.0) {
	// Only report a problem if the difference is significant.
	if (-delay1 > vth_time_tol * vo_delay_)
	  fail("load delay less than zero");
	// Use elmore delay.
	delay1 = elmore;
      }
      if (slew1 < drvr_slew_) {
	// Only report a problem if the difference is significant.
	if ((drvr_slew_ - slew1) > vth_time_tol * drvr_slew_)
	  fail("load slew less than driver slew");
	slew1 = drvr_slew_;
      }
      delay = delay1;
      slew = slew1;
    }
    catch (DmpError &error) {
      fail(error.what());
      delay = elmore_;
      slew = drvr_slew_;
    }
  }
}

// Find t such that vl(t)=v.
double
DmpAlg::findVlCrossing(double vth,
                       double t_lower,
                       double t_upper)
{
  FindRootFunc vl_func = [&] (double t,
                              double &y,
                              double &dy) {
    double vl, vl_dt;
    Vl(t, vl, vl_dt);
    y = vl - vth;
    dy = vl_dt;
  };
  bool fail;
  double t_vth = findRoot(vl_func, t_lower, t_upper, vth_time_tol,
                          find_root_max_iter, fail);
  if (fail)
    throw DmpError("find Vl crossing failed");
  return t_vth;
}

double
DmpAlg::vlCrossingUpperBound()
{
  return voCrossingUpperBound() + elmore_ * 2.0;
}

void
DmpAlg::Vl(double t,
           // Return values.
           double &vl,
           double &dvl_dt)
{
  double t1 = t - t0_;
  if (t1 <= 0.0) {
    vl = 0.0;
    dvl_dt = 0.0;
  }
  else if (t1 <= dt_) {
    double vl0, dvl0_dt;
    Vl0(t1, vl0, dvl0_dt);
    vl = vl0 / dt_;
    dvl_dt = dvl0_dt / dt_;
  }
  else {
    double vl0, dvl0_dt;
    Vl0(t1, vl0, dvl0_dt);

    double vl0_dt, dvl0_dt_dt;
    Vl0(t1 - dt_, vl0_dt, dvl0_dt_dt);

    vl = (vl0 - vl0_dt) / dt_;
    dvl_dt = (dvl0_dt - dvl0_dt_dt) / dt_;
  }
}

void
DmpAlg::showVl()
{
  report_->reportLine("  t    vl(t)");
  double ub = vlCrossingUpperBound();
  for (double t = t0_; t < t0_ + ub * 2.0; t += ub / 10.0) {
    double vl, dvl_dt;
    Vl(t, vl, dvl_dt);
    report_->reportLine(" %g %g", t, vl);
  }
}

void
DmpAlg::fail(const char *reason)
{
  // Report failures with a unique debug flag.
  if (debug_->check("dmp_ceff", 1) || debug_->check("dcalc_error", 1))
    report_->reportLine("delay_calc: DMP failed - %s c2=%s rpi=%s c1=%s rd=%s",
                        reason,
                        units_->capacitanceUnit()->asString(c2_),
                        units_->resistanceUnit()->asString(rpi_),
                        units_->capacitanceUnit()->asString(c1_),
                        units_->resistanceUnit()->asString(rd_));
}

////////////////////////////////////////////////////////////////

// Capacitive load.
class DmpCap : public DmpAlg
{
public:
  DmpCap(StaState *sta);
  const char *name() override { return "cap"; }
  void init(const LibertyLibrary *library,
            const LibertyCell *drvr_cell,
            const Pvt *pvt,
            const GateTableModel *gate_model,
            const RiseFall *rf,
            double rd,
            double in_slew,
            double c2,
            double rpi,
            double c1) override;
  void gateDelaySlew(// Return values.
                     double &delay,
                     double &slew) override;
  void loadDelaySlew(const Pin *,
                     double elmore,
                     // Return values.
                     ArcDelay &delay,
                     Slew &slew) override;
  void evalDmpEqns() override;
  double voCrossingUpperBound() override;

private:
  void V0(double t,
          // Return values.
          double &vo,
          double &dvo_dt) override;
  void Vl0(double t,
           // Return values.
           double &vl,
           double &dvl_dt) override;
};

DmpCap::DmpCap(StaState *sta):
  DmpAlg(1, sta)
{
}

void
DmpCap::init(const LibertyLibrary *drvr_library,
	     const LibertyCell *drvr_cell,
	     const Pvt *pvt,
	     const GateTableModel *gate_model,
	     const RiseFall *rf,
	     double rd,
	     double in_slew,
	     double c2,
	     double rpi,
	     double c1)
{
  debugPrint(debug_, "dmp_ceff", 3, "Using DMP cap");
  DmpAlg::init(drvr_library, drvr_cell, pvt, gate_model, rf,
	       rd, in_slew, c2, rpi, c1);
  ceff_ = c1 + c2;
}

void
DmpCap::gateDelaySlew(// Return values.
                      double &delay,
		      double &slew)
{
  debugPrint(debug_, "dmp_ceff", 3, "    ceff = %s",
             units_->capacitanceUnit()->asString(ceff_));
  gateCapDelaySlew(ceff_, delay, slew);
  drvr_slew_ = slew;
}

void
DmpCap::loadDelaySlew(const Pin *,
		      double elmore,
		      ArcDelay &delay,
		      Slew &slew)
{
  delay = elmore;
  slew = drvr_slew_;
}

void
DmpCap::evalDmpEqns()
{
}

void
DmpCap::V0(double,
           // Return values.
           double &vo,
           double &dvo_dt)
{
  vo = 0.0;
  dvo_dt = 0.0;
}

double
DmpCap::voCrossingUpperBound()
{
  return 0.0;
}

void
DmpCap::Vl0(double ,
            // Return values.
            double &vl,
            double &dvl_dt)
{
  vl = 0.0;
  dvl_dt = 0.0;
}

////////////////////////////////////////////////////////////////

// No non-zero pi model parameters, two poles, one zero
class DmpPi : public DmpAlg
{
public:
  DmpPi(StaState *sta);
  const char *name() override { return "Pi"; }
  void init(const LibertyLibrary *library,
            const LibertyCell *drvr_cell,
            const Pvt *pvt,
            const GateTableModel *gate_model,
            const RiseFall *rf,
            double rd,
            double in_slew,
            double c2,
            double rpi,
            double c1) override;
  void gateDelaySlew(// Return values.
                     double &delay,
                     double &slew) override;
  void evalDmpEqns() override;
  double voCrossingUpperBound() override;

private:
  void findDriverParamsPi();
  double ipiIceff(double t0,
		  double dt,
		  double ceff_time,
		  double ceff);
  void V0(double t,
          // Return values.
          double &vo,
          double &dvo_dt) override;
  void Vl0(double t,
           // Return values.
           double &vl,
           double &dvl_dt) override;

  // Poles/zero.
  double p1_;
  double p2_;
  double z1_;
  // Residues.
  double k0_;
  double k1_;
  double k2_;
  double k3_;
  double k4_;
  // Ipi coefficients.
  double A_;
  double B_;
  double D_;
};

DmpPi::DmpPi(StaState *sta) :
  DmpAlg(3, sta),
  p1_(0.0),
  p2_(0.0),
  z1_(0.0),
  k0_(0.0),
  k1_(0.0),
  k2_(0.0),
  k3_(0.0),
  k4_(0.0),
  A_(0.0),
  B_(0.0),
  D_(0.0)
{
}

void
DmpPi::init(const LibertyLibrary *drvr_library,
	    const LibertyCell *drvr_cell,
	    const Pvt *pvt,
	    const GateTableModel *gate_model,
	    const RiseFall *rf,
	    double rd,
	    double in_slew,
	    double c2,
	    double rpi,
	    double c1)
{
  debugPrint(debug_, "dmp_ceff", 3, "Using DMP Pi");
  DmpAlg::init(drvr_library, drvr_cell, pvt, gate_model, rf, rd,
	       in_slew, c2, rpi, c1);

  // Find poles/zeros.
  z1_ = 1.0 / (rpi_ * c1_);
  k0_ = 1.0 / (rd_ * c2_);
  double a = rpi_ * rd_ * c1_ * c2_;
  double b = rd_ * (c1_ + c2_) + rpi_ * c1_;
  double sqrt_ = sqrt(b * b - 4 * a);
  p1_ = (b + sqrt_) / (2 * a);
  p2_ = (b - sqrt_) / (2 * a);

  double p1p2 = (p1_ * p2_);
  k2_ = z1_ / p1p2;
  k1_ = (1.0 - k2_ * (p1_ + p2_)) / p1p2;
  k4_ = (k1_ * p1_ + k2_) / (p2_ - p1_);
  k3_ = -k1_ - k4_;

  double z_ = (c1_ + c2_) / (rpi_ * c1_ * c2_);
  A_ = z_ / p1p2;
  B_ = (z_ - p1_) / (p1_ * (p1_ - p2_));
  D_ = (z_ - p2_) / (p2_ * (p2_ - p1_));
}

void
DmpPi::gateDelaySlew(// Return values.
                     double &delay,
		     double &slew)
{
  driver_valid_ = false;
  try {
    findDriverParamsPi();
    ceff_ = x_[DmpParam::ceff];
    double table_delay, table_slew;
    gateCapDelaySlew(ceff_, table_delay, table_slew);
    delay = table_delay;
    //slew = table_slew;
    try {
      double vo_delay, vo_slew;
      findDriverDelaySlew(vo_delay, vo_slew);
      driver_valid_ = true;
      // Save Vo delay to measure load wire delay waveform.
      vo_delay_ = vo_delay;
      //delay = vo_delay;
      slew = vo_slew;
    }
    catch (DmpError &error) {
      fail(error.what());
      // Fall back to table slew.
      slew = table_slew;
    }
  }
  catch (DmpError &error) {
    fail(error.what());
    // Driver calculation failed - use Ceff=c1+c2.
    ceff_ = c1_ + c2_;
    gateCapDelaySlew(ceff_, delay, slew);
  }
  drvr_slew_ = slew;
}

void
DmpPi::findDriverParamsPi()
{
  try {
    findDriverParams(c2_ + c1_);
  }
  catch (DmpError &) {
    findDriverParams(c2_);
  }
}

// Given x_ as a vector of input parameters, fill fvec_ with the
// equations evaluated at x_ and fjac_ with the jacobian evaluated at x_.
void
DmpPi::evalDmpEqns()
{
  double t0 = x_[DmpParam::t0];
  double dt = x_[DmpParam::dt];
  double ceff = x_[DmpParam::ceff];

  if (ceff < 0.0)
    throw DmpError("eqn eval failed: ceff < 0");
  if (ceff > (c1_ + c2_))
    throw DmpError("eqn eval failed: ceff > c2 + c1");

  double t_vth, t_vl, slew;
  gateDelays(ceff, t_vth, t_vl, slew);
  if (slew == 0.0)
    throw DmpError("eqn eval failed: slew = 0");

  double ceff_time = slew / (vh_ - vl_);
  if (ceff_time > 1.4 * dt)
    ceff_time = 1.4 * dt;

  if (dt <= 0.0)
    throw DmpError("eqn eval failed: dt < 0");

  double exp_p1_dt = exp2(-p1_ * dt);
  double exp_p2_dt = exp2(-p2_ * dt);
  double exp_dt_rd_ceff = exp2(-dt / (rd_ * ceff));

  double y50 = y(t_vth, t0, dt, ceff);
  // Match Vl.
  double y20 = y(t_vl, t0, dt, ceff);
  fvec_[DmpFunc::ipi] = ipiIceff(t0, dt, ceff_time, ceff);
  fvec_[DmpFunc::y50] = y50 - vth_;
  fvec_[DmpFunc::y20] = y20 - vl_;
  fjac_[DmpFunc::ipi][DmpParam::t0] = 0.0;
  fjac_[DmpFunc::ipi][DmpParam::dt] =
    (-A_ * dt + B_ * dt * exp_p1_dt - (2 * B_ / p1_) * (1.0 - exp_p1_dt)
     + D_ * dt * exp_p2_dt - (2 * D_ / p2_) * (1.0 - exp_p2_dt)
     + rd_ * ceff * (dt + dt * exp_dt_rd_ceff
		     - 2 * rd_ * ceff * (1.0 - exp_dt_rd_ceff)))
    / (rd_ * dt * dt * dt);
  fjac_[DmpFunc::ipi][DmpParam::ceff] =
    (2 * rd_ * ceff - dt - (2 * rd_ * ceff + dt) * exp2(-dt / (rd_ * ceff)))
    / (dt * dt);

  dy(t_vl, t0, dt, ceff,
     fjac_[DmpFunc::y20][DmpParam::t0],
     fjac_[DmpFunc::y20][DmpParam::dt],
     fjac_[DmpFunc::y20][DmpParam::ceff]);

  dy(t_vth, t0, dt, ceff,
     fjac_[DmpFunc::y50][DmpParam::t0],
     fjac_[DmpFunc::y50][DmpParam::dt],
     fjac_[DmpFunc::y50][DmpParam::ceff]);

  if (debug_->check("dmp_ceff", 4)) {
    showX();
    showFvec();
    showJacobian();
    report_->reportLine(".................");
  }
}

// Eqn 13, Eqn 14.
double
DmpPi::ipiIceff(double, double dt,
		double ceff_time,
		double ceff)
{
  double exp_p1_dt = exp2(-p1_ * ceff_time);
  double exp_p2_dt = exp2(-p2_ * ceff_time);
  double exp_dt_rd_ceff = exp2(-ceff_time / (rd_ * ceff));
  double ipi = (A_ * ceff_time + (B_ / p1_) * (1.0 - exp_p1_dt)
	       + (D_ / p2_) * (1.0 - exp_p2_dt))
    / (rd_ * ceff_time * dt);
  double iceff = (rd_ * ceff * ceff_time - (rd_ * ceff) * (rd_ * ceff)
		  * (1.0 - exp_dt_rd_ceff))
    / (rd_ * ceff_time * dt);
  return ipi - iceff;
}

void
DmpPi::V0(double t,
          // Return values.
          double &vo,
          double &dvo_dt)
{
  double exp_p1 = exp2(-p1_ * t);
  double exp_p2 = exp2(-p2_ * t);
  vo = k0_ * (k1_ + k2_ * t + k3_ * exp_p1 + k4_ * exp_p2);
  dvo_dt = k0_ * (k2_ - k3_ * p1_ * exp_p1 - k4_ * p2_ * exp_p2);
}

void
DmpPi::Vl0(double t,
           // Return values.
           double &vl,
           double &dvl_dt)
{
  double D1 = k0_ * (k1_ - k2_ / p3_);
  double D3 = -p3_ * k0_ * k3_ / (p1_ - p3_);
  double D4 = -p3_ * k0_ * k4_ / (p2_ - p3_);
  double D5 = k0_ * (k2_ / p3_ - k1_ + p3_ * k3_ / (p1_ - p3_)
		    + p3_ * k4_ / (p2_ - p3_));
  double exp_p1 = exp2(-p1_ * t);
  double exp_p2 = exp2(-p2_ * t);
  double exp_p3 = exp2(-p3_ * t);
  vl = D1 + t + D3 * exp_p1 + D4 * exp_p2 + D5 * exp_p3;
  dvl_dt = 1.0 - D3 * p1_ * exp_p1 - D4 * p2_ * exp_p2
    - D5 * p3_ * exp_p3;
}

double
DmpPi::voCrossingUpperBound()
{
  return t0_ + dt_ + (c1_ + c2_) * (rd_ + rpi_) * 2.0;
}

////////////////////////////////////////////////////////////////

// Capacitive load, so Ceff is known.
// Solve for t0, delta t.
class DmpOnePole : public DmpAlg
{
public:
  DmpOnePole(StaState *sta);
  void evalDmpEqns() override;
  double voCrossingUpperBound() override;
};

DmpOnePole::DmpOnePole(StaState *sta) :
  DmpAlg(2, sta)
{
}

void
DmpOnePole::evalDmpEqns()
{
  double t0 = x_[DmpParam::t0];
  double dt = x_[DmpParam::dt];

  double t_vth, t_vl, ignore1, ignore2;
  gateDelays(ceff_, t_vth, t_vl, ignore1);

  if (dt <= 0.0)
    dt = x_[DmpParam::dt] = (t_vl - t_vth) / 100;

  fvec_[DmpFunc::y50] = y(t_vth, t0, dt, ceff_) - vth_;
  fvec_[DmpFunc::y20] = y(t_vl, t0, dt, ceff_) - vl_;

  if (debug_->check("dmp_ceff", 4)) {
    showX();
    showFvec();
  }

  dy(t_vl, t0, dt, ceff_,
     fjac_[DmpFunc::y20][DmpParam::t0],
     fjac_[DmpFunc::y20][DmpParam::dt],
     ignore2);

  dy(t_vth, t0, dt, ceff_,
     fjac_[DmpFunc::y50][DmpParam::t0],
     fjac_[DmpFunc::y50][DmpParam::dt],
     ignore2);

  if (debug_->check("dmp_ceff", 4)) {
    showJacobian();
    report_->reportLine(".................");
  }
}

double
DmpOnePole::voCrossingUpperBound()
{
  return t0_ + dt_ + ceff_ * rd_ * 2.0;
}

////////////////////////////////////////////////////////////////

// C2 = 0, one pole, one zero.
class DmpZeroC2 : public DmpOnePole
{
public:
  DmpZeroC2(StaState *sta);
  const char *name() override { return "c2=0"; }
  void init(const LibertyLibrary *drvr_library,
            const LibertyCell *drvr_cell,
            const Pvt *pvt,
            const GateTableModel *gate_model,
            const RiseFall *rf,
            double rd,
            double in_slew,
            double c2,
            double rpi,
            double c1) override;
  void gateDelaySlew(// Return values.
                     double &delay,
                     double &slew) override;

private:
  void V0(double t,
          // Return values.
          double &vo,
          double &dvo_dt) override;
   void Vl0(double t,
            // Return values.
            double &vl,
            double &dvl_dt) override;
  double voCrossingUpperBound() override;

  // Pole/zero.
  double p1_;
  double z1_;
  // Residues.
  double k0_;
  double k1_;
  double k2_;
  double k3_;
};

DmpZeroC2::DmpZeroC2(StaState *sta) :
  DmpOnePole(sta),
  p1_(0.0),
  z1_(0.0),
  k0_(0.0),
  k1_(0.0),
  k2_(0.0),
  k3_(0.0)
{
}

void
DmpZeroC2::init(const LibertyLibrary *drvr_library,
		const LibertyCell *drvr_cell,
		const Pvt *pvt,
		const GateTableModel *gate_model,
		const RiseFall *rf,
		double rd,
		double in_slew,
		double c2,
		double rpi,
		double c1)
{
  debugPrint(debug_, "dmp_ceff", 3, "Using DMP C2=0");
  DmpAlg::init(drvr_library, drvr_cell, pvt, gate_model, rf, rd,
	       in_slew, c2, rpi, c1);
  ceff_ = c1;

  z1_ = 1.0 / (rpi_ * c1_);
  p1_ = 1.0 / (c1_ * (rd_ + rpi_));

  k0_ = p1_ / z1_;
  k2_ = 1.0 / k0_;
  k1_ = (p1_ - z1_) / (p1_ * p1_);
  k3_ = -k1_;
}

void
DmpZeroC2::gateDelaySlew(// Return values.
                         double &delay,
			 double &slew)
{
  try {
    findDriverParams(c1_);
    ceff_ = c1_;
    findDriverDelaySlew(delay, slew);
    driver_valid_ = true;
    vo_delay_ = delay;
  }
  catch (DmpError &error) {
    fail(error.what());
    // Fall back to table slew.
    driver_valid_ = false;
    ceff_ = c1_;
    gateCapDelaySlew(ceff_, delay, slew);
  }
  drvr_slew_ = slew;
}

void
DmpZeroC2::V0(double t,
              // Return values.
              double &vo,
              double &dvo_dt)
{
  double exp_p1 = exp2(-p1_ * t);
  vo = k0_ * (k1_ + k2_ * t + k3_ * exp_p1);
  dvo_dt = k0_ * (k2_ - k3_ * p1_ * exp_p1);
}

void
DmpZeroC2::Vl0(double t,
           // Return values.
           double &vl,
           double &dvl_dt)
{
  double D1 = k0_ * (k1_ - k2_ / p3_);
  double D3 = -p3_ * k0_ * k3_ / (p1_ - p3_);
  double D5 = k0_ * (k2_ / p3_ - k1_ + p3_ * k3_ / (p1_ - p3_));
  double exp_p1 = exp2(-p1_ * t);
  double exp_p3 = exp2(-p3_ * t);
  vl = D1 + t + D3 * exp_p1 + D5 * exp_p3;
  dvl_dt = 1.0 - D3 * p1_ * exp_p1 - D5 * p3_ * exp_p3;
}

double
DmpZeroC2::voCrossingUpperBound()
{
  return t0_ + dt_ + c1_ * (rd_ + rpi_) * 2.0;
}

////////////////////////////////////////////////////////////////

// Newton-Raphson iteration to find zeros of a function.
// x_tol is percentage that all changes in x must be less than (1.0 = 100%).
// Eval(state) is called to fill fvec and fjac (returns false if fails).
// Return error msg on failure.
static void
newtonRaphson(const int max_iter,
	      double x[],
	      const int size,
	      const double x_tol,
	      function<void ()> eval,
	      // Temporaries supplied by caller.
	      double *fvec,
	      double **fjac,
	      int *index,
	      double *p,
	      double *scale)
{
  for (int k = 0; k < max_iter; k++) {
    eval();
    for (int i = 0; i < size; i++)
      // Right-hand side of linear equations.
      p[i] = -fvec[i];
    luDecomp(fjac, size, index, scale);
    luSolve(fjac, size, index, p);

    bool all_under_x_tol = true;
    for (int i = 0; i < size; i++) {
      if (abs(p[i]) > abs(x[i]) * x_tol)
	all_under_x_tol = false;
      x[i] += p[i];
    }
    if (all_under_x_tol)
      return;
  }
  throw DmpError("Newton-Raphson max iterations exceeded");
}

// luDecomp, luSolve based on MatClass from C. R. Birchenhall,
// University of Manchester
// ftp://ftp.mcc.ac.uk/pub/matclass/libmat.tar.Z

// Crout's Method of LU decomposition of square matrix, with implicit
// partial pivoting.  A is overwritten. U is explicit in the upper
// triangle and L is in multiplier form in the subdiagionals i.e. subdiag
// a[i,j] is the multiplier used to eliminate the [i,j] term.
//
// Replaces a[0..size-1][0..size-1] by the LU decomposition.
// index[0..size-1] is an output vector of the row permutations.
// Return error msg on failure.
void
luDecomp(double **a,
	 const int size,
	 int *index,
	 // Temporary supplied by caller.
	 // scale stores the implicit scaling of each row.
	 double *scale)
{
  // Find implicit scaling factors.
  for (int i = 0; i < size; i++) {
    double big = 0.0;
    for (int j = 0; j < size; j++) {
      double temp = abs(a[i][j]);
      if (temp > big)
	big = temp;
    }
    if (big == 0.0)
      throw DmpError("LU decomposition: no non-zero row element");
    scale[i] = 1.0 / big;
  }
  int size_1 = size - 1;
  for (int j = 0; j < size; j++) {
    // Run down jth column from top to diag, to form the elements of U.
    for (int i = 0; i < j; i++) {
      double sum = a[i][j];
      for (int k = 0; k < i; k++)
	sum -= a[i][k] * a[k][j];
      a[i][j] = sum;
    }
    // Run down jth subdiag to form the residuals after the elimination
    // of the first j-1 subdiags.  These residuals diviyded by the
    // appropriate diagonal term will become the multipliers in the
    // elimination of the jth. subdiag. Find index of largest scaled
    // term in imax.
    double big = 0.0;
    int imax = 0;
    for (int i = j; i < size; i++) {
      double sum = a[i][j];
      for (int k = 0; k < j; k++)
	sum -= a[i][k] * a[k][j];
      a[i][j] = sum;
      double dum = scale[i] * abs(sum);
      if (dum >= big) {
	big = dum;
	imax = i;
      }
    }
    // Permute current row with imax.
    if (j != imax) {
      // Yes, do so...
      for (int k = 0; k < size; k++) {
	double dum = a[imax][k];
	a[imax][k] = a[j][k];
	a[j][k] = dum;
      }
      scale[imax] = scale[j];
    }
    index[j] = imax;
    // If diag term is not zero divide subdiag to form multipliers.
    if (a[j][j] == 0.0)
      a[j][j] = tiny_double;
    if (j != size_1) {
      double pivot = 1.0 / a[j][j];
      for (int i = j + 1; i < size; i++)
	a[i][j] *= pivot;
    }
  }
}

// Solves the set of size linear equations a*x=b, assuming A is LU form
// but assume b has not been transformed.
//  a[0..size-1] is LU decomposition
// Returns the solution vector x in b.
// a and index are not modified.
void
luSolve(double **a,
	const int size,
	const int *index,
	double b[])
{
  // Transform b allowing for leading zeros.
  int non_zero = -1;
  for (int i = 0; i < size; i++) {
    int iperm = index[i];
    double sum = b[iperm];
    b[iperm] = b[i];
    if (non_zero != -1) {
      for (int j = non_zero; j <= i - 1; j++)
	sum -= a[i][j] * b[j];
    }
    else {
      if (sum != 0.0)
	non_zero = i;
    }
    b[i] = sum;
  }
  // Backsubstitution.
  for (int i = size - 1; i >= 0; i--) {
    double sum = b[i];
    for (int j = i + 1; j < size; j++)
      sum -= a[i][j] * b[j];
    b[i] = sum / a[i][i];
  }
}

#if 0
// Solve:
//  x + y = 5
//  x - y = 1
// x = 3
// y = 2
void
testLuDecomp1()
{
  double a0[2] = {1,  1};
  double a1[2] = {1, -1};
  double *a[2] = {a0, a1};
  int index[2];
  double b[2] = {5, 1};
  double scale[2];
  luDecomp(a, 2, index, scale);
  luSolve(a, 2, index, b);
  printf("x = %f y= %f\n", b[0], b[1]);
}

// Solve
//   x + 2y =  3
//  3x - 4y = 19
// x = 5
// y = -1
void
testLuDecomp2()
{
  double a0[2] = {1,  2};
  double a1[2] = {3, -4};
  double *a[2] = {a0, a1};
  int index[2];
  double b[2] = {3, 19};
  double scale[2];
  luDecomp(a, 2, index, scale);
  luSolve(a, 2, index, b);
  printf("x = %f y= %f\n", b[0], b[1]);
}
#endif

////////////////////////////////////////////////////////////////

bool DmpCeffDelayCalc::unsuppored_model_warned_ = false;

DmpCeffDelayCalc::DmpCeffDelayCalc(StaState *sta) :
  LumpedCapDelayCalc(sta),
  dmp_cap_(new DmpCap(sta)),
  dmp_pi_(new DmpPi(sta)),
  dmp_zero_c2_(new DmpZeroC2(sta)),
  dmp_alg_(nullptr)
{
}

DmpCeffDelayCalc::~DmpCeffDelayCalc()
{
  delete dmp_cap_;
  delete dmp_pi_;
  delete dmp_zero_c2_;
}

ArcDcalcResult
DmpCeffDelayCalc::gateDelay(const Pin *drvr_pin,
                            const TimingArc *arc,
                            const Slew &in_slew,
                            float load_cap,
                            const Parasitic *parasitic,
                            const LoadPinIndexMap &load_pin_index_map,
                            const DcalcAnalysisPt *dcalc_ap)
{
  const RiseFall *rf = arc->toEdge()->asRiseFall();
  const LibertyCell *drvr_cell = arc->from()->libertyCell();
  const LibertyLibrary *drvr_library = drvr_cell->libertyLibrary();

  GateTableModel *table_model = arc->gateTableModel(dcalc_ap);
  if (table_model && parasitic) {
    float in_slew1 = delayAsFloat(in_slew);
    float c2, rpi, c1;
    parasitics_->piModel(parasitic, c2, rpi, c1);
    if (isnan(c2) || isnan(c1) || isnan(rpi))
      report_->error(1040, "parasitic Pi model has NaNs.");
    setCeffAlgorithm(drvr_library, drvr_cell, pinPvt(drvr_pin, dcalc_ap),
                     table_model, rf, in_slew1, c2, rpi, c1);
    double gate_delay, drvr_slew;
    gateDelaySlew(gate_delay, drvr_slew);
    ArcDcalcResult dcalc_result(load_pin_index_map.size());
    dcalc_result.setGateDelay(gate_delay);
    dcalc_result.setDrvrSlew(drvr_slew);

    for (const auto &[load_pin, load_idx] : load_pin_index_map) {
      ArcDelay wire_delay;
      Slew load_slew;
      loadDelaySlew(load_pin, drvr_slew, rf, drvr_library, parasitic,
                    wire_delay, load_slew);
      dcalc_result.setWireDelay(load_idx, wire_delay);
      dcalc_result.setLoadSlew(load_idx, load_slew);
    }
    return dcalc_result;
  }
  else {
    ArcDcalcResult dcalc_result =
      LumpedCapDelayCalc::gateDelay(drvr_pin, arc, in_slew, load_cap, parasitic,
                                    load_pin_index_map, dcalc_ap);
    if (parasitic
	&& !unsuppored_model_warned_) {
      unsuppored_model_warned_ = true;
      report_->warn(1041, "cell %s delay model not supported on SPF parasitics by DMP delay calculator",
		    drvr_cell->name());
    }
    return dcalc_result;
  }
}

void
DmpCeffDelayCalc::setCeffAlgorithm(const LibertyLibrary *drvr_library,
				   const LibertyCell *drvr_cell,
				   const Pvt *pvt,
				   const GateTableModel *gate_model,
				   const RiseFall *rf,
				   double in_slew,
				   double c2,
				   double rpi,
				   double c1)
{
  double rd = 0.0;
  if (gate_model) {
    rd = gateModelRd(drvr_cell, gate_model, rf, in_slew, c2, c1,
		     pvt, variables_->pocvEnabled());
    // Zero Rd means the table is constant and thus independent of load cap.
    if (rd < 1e-2
	// Rpi is small compared to Rd, which makes the load capacitive.
	|| rpi < rd * 1e-3
	// c1/Rpi can be ignored.
	|| (c1 == 0.0 || c1 < c2 * 1e-3 || rpi == 0.0))
      dmp_alg_ = dmp_cap_;
    else if (c2 < c1 * 1e-3)
      dmp_alg_ = dmp_zero_c2_;
    else
      // The full monty.
      dmp_alg_ = dmp_pi_;
  }
  else
    dmp_alg_ = dmp_cap_;
  dmp_alg_->init(drvr_library, drvr_cell, pvt, gate_model,
		 rf, rd, in_slew, c2, rpi, c1);
  debugPrint(debug_, "dmp_ceff", 3,
             "    DMP in_slew = %s c2 = %s rpi = %s c1 = %s Rd = %s (%s alg)",
             units_->timeUnit()->asString(in_slew),
             units_->capacitanceUnit()->asString(c2),
             units_->resistanceUnit()->asString(rpi),
             units_->capacitanceUnit()->asString(c1),
             units_->resistanceUnit()->asString(rd),
             dmp_alg_->name());
}

string
DmpCeffDelayCalc::reportGateDelay(const Pin *drvr_pin,
                                  const TimingArc *arc,
				  const Slew &in_slew,
				  float load_cap,
				  const Parasitic *parasitic,
                                  const LoadPinIndexMap &load_pin_index_map,
				  const DcalcAnalysisPt *dcalc_ap,
				  int digits)
{
  ArcDcalcResult dcalc_result = gateDelay(drvr_pin, arc, in_slew, load_cap,
                                          parasitic, load_pin_index_map, dcalc_ap);
  GateTableModel *model = arc->gateTableModel(dcalc_ap);
  float c_eff = 0.0;
  string result;
  const LibertyCell *drvr_cell = arc->to()->libertyCell();
  const LibertyLibrary *drvr_library = drvr_cell->libertyLibrary();
  const Units *units = drvr_library->units();
  const Unit *cap_unit = units->capacitanceUnit();
  const Unit *res_unit = units->resistanceUnit();
  if (parasitic && dmp_alg_) {
    c_eff = dmp_alg_->ceff();
    float c2, rpi, c1;
    parasitics_->piModel(parasitic, c2, rpi, c1);
    result += "Pi model C2=";
    result += cap_unit->asString(c2, digits);
    result += " Rpi=";
    result += res_unit->asString(rpi, digits);
    result += " C1=";
    result += cap_unit->asString(c1, digits);
    result += ", Ceff=";
    result += cap_unit->asString(c_eff, digits);
    result += '\n';
  }
  else
    c_eff = load_cap;
  if (model) {
    const Unit *time_unit = units->timeUnit();
    float in_slew1 = delayAsFloat(in_slew);
    result += model->reportGateDelay(pinPvt(drvr_pin, dcalc_ap), in_slew1, c_eff,
                                     variables_->pocvEnabled(), digits);
    result += "Driver waveform slew = ";
    float drvr_slew = delayAsFloat(dcalc_result.drvrSlew());
    result += time_unit->asString(drvr_slew, digits);
    result += '\n';
  }
  return result;
}

static double
gateModelRd(const LibertyCell *cell,
	    const GateTableModel *gate_model,
	    const RiseFall *rf,
	    double in_slew,
	    double c2,
	    double c1,
	    const Pvt *pvt,
	    bool pocv_enabled)
{
  float cap1 = c1 + c2;
  float cap2 = cap1 + 1e-15;
  ArcDelay d1, d2;
  Slew s1, s2;
  gate_model->gateDelay(pvt, in_slew, cap1, pocv_enabled, d1, s1);
  gate_model->gateDelay(pvt, in_slew, cap2, pocv_enabled, d2, s2);
  double vth = cell->libertyLibrary()->outputThreshold(rf);
  float rd = -log(vth) * abs(delayAsFloat(d1) - delayAsFloat(d2)) / (cap2 - cap1);
  return rd;
}

void
DmpCeffDelayCalc::gateDelaySlew(// Return values.
                                double &delay,
				double &slew)
{
  dmp_alg_->gateDelaySlew(delay, slew);
}

void
DmpCeffDelayCalc::loadDelaySlewElmore(const Pin *load_pin,
                                      double elmore,
                                      ArcDelay &delay,
                                      Slew &slew)
{
  if (dmp_alg_)
    dmp_alg_->loadDelaySlew(load_pin, elmore, delay, slew);
}

// Notify algorithm components.
void
DmpCeffDelayCalc::copyState(const StaState *sta)
{
  StaState::copyState(sta);
  dmp_cap_->copyState(sta);
  dmp_pi_->copyState(sta);
  dmp_zero_c2_->copyState(sta);
}

DmpError::DmpError(const char *what) :
  what_(what)
{
  //printf("DmpError %s\n", what);
}

// This saves about 2.5% in overall run time on designs with SPEF.
// https://codingforspeed.com/using-faster-exponential-approximation
static double
exp2(double x)
{
  if (x < -12.0)
    // exp(-12) = 6.1e-6
    return 0.0;
  else {
    double y = 1.0 + x / 4096.0;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    y *= y;
    return y;
  }
}

} // namespace