Happy New Year!
[ginac.git] / ginac / inifcns.cpp
index 17a080c..e2a09b1 100644 (file)
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's initially known functions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
  *
  *  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
  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
-#include <vector>
-#include <stdexcept>
-
 #include "inifcns.h"
 #include "ex.h"
 #include "constant.h"
 #include "lst.h"
+#include "fderivative.h"
 #include "matrix.h"
 #include "mul.h"
 #include "power.h"
@@ -37,6 +35,9 @@
 #include "symmetry.h"
 #include "utils.h"
 
+#include <stdexcept>
+#include <vector>
+
 namespace GiNaC {
 
 //////////
@@ -66,6 +67,19 @@ static ex conjugate_conjugate(const ex & arg)
        return arg;
 }
 
+// If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x) 
+// and conjugate((U+I*V).diff(x))
+static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return conjugate(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
 static ex conjugate_real_part(const ex & arg)
 {
        return arg.real_part();
@@ -76,8 +90,46 @@ static ex conjugate_imag_part(const ex & arg)
        return -arg.imag_part();
 }
 
+static bool func_arg_info(const ex & arg, unsigned inf)
+{
+       // for some functions we can return the info() of its argument
+       // (think of conjugate())
+       switch (inf) {
+               case info_flags::polynomial:
+               case info_flags::integer_polynomial:
+               case info_flags::cinteger_polynomial:
+               case info_flags::rational_polynomial:
+               case info_flags::real:
+               case info_flags::rational:
+               case info_flags::integer:
+               case info_flags::crational:
+               case info_flags::cinteger:
+               case info_flags::even:
+               case info_flags::odd:
+               case info_flags::prime:
+               case info_flags::crational_polynomial:
+               case info_flags::rational_function:
+               case info_flags::positive:
+               case info_flags::negative:
+               case info_flags::nonnegative:
+               case info_flags::posint:
+               case info_flags::negint:
+               case info_flags::nonnegint:
+               case info_flags::has_indices:
+                       return arg.info(inf);
+       }
+       return false;
+}
+
+static bool conjugate_info(const ex & arg, unsigned inf)
+{
+       return func_arg_info(arg, inf);
+}
+
 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
                                       evalf_func(conjugate_evalf).
+                                      expl_derivative_func(conjugate_expl_derivative).
+                                      info_func(conjugate_info).
                                       print_func<print_latex>(conjugate_print_latex).
                                       conjugate_func(conjugate_conjugate).
                                       real_part_func(conjugate_real_part).
@@ -121,8 +173,21 @@ static ex real_part_imag_part(const ex & arg)
        return 0;
 }
 
+// If x is real then Re(e).diff(x) is equal to Re(e.diff(x)) 
+static ex real_part_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return real_part_function(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
                                       evalf_func(real_part_evalf).
+                                      expl_derivative_func(real_part_expl_derivative).
                                       print_func<print_latex>(real_part_print_latex).
                                       conjugate_func(real_part_conjugate).
                                       real_part_func(real_part_real_part).
@@ -166,8 +231,21 @@ static ex imag_part_imag_part(const ex & arg)
        return 0;
 }
 
+// If x is real then Im(e).diff(x) is equal to Im(e.diff(x)) 
+static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return imag_part_function(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
                                       evalf_func(imag_part_evalf).
+                                      expl_derivative_func(imag_part_expl_derivative).
                                       print_func<print_latex>(imag_part_print_latex).
                                       conjugate_func(imag_part_conjugate).
                                       real_part_func(imag_part_real_part).
@@ -194,12 +272,58 @@ static ex abs_eval(const ex & arg)
        if (arg.info(info_flags::nonnegative))
                return arg;
 
+       if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
+               return -arg;
+
        if (is_ex_the_function(arg, abs))
                return arg;
 
+       if (is_ex_the_function(arg, exp))
+               return exp(arg.op(0).real_part());
+
+       if (is_exactly_a<power>(arg)) {
+               const ex& base = arg.op(0);
+               const ex& exponent = arg.op(1);
+               if (base.info(info_flags::positive) || exponent.info(info_flags::real))
+                       return pow(abs(base), exponent.real_part());
+       }
+
+       if (is_ex_the_function(arg, conjugate_function))
+               return abs(arg.op(0));
+
+       if (is_ex_the_function(arg, step))
+               return arg;
+
        return abs(arg).hold();
 }
 
+static ex abs_expand(const ex & arg, unsigned options)
+{
+       if ((options & expand_options::expand_transcendental)
+               && is_exactly_a<mul>(arg)) {
+               exvector prodseq;
+               prodseq.reserve(arg.nops());
+               for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
+                       if (options & expand_options::expand_function_args)
+                               prodseq.push_back(abs(i->expand(options)));
+                       else
+                               prodseq.push_back(abs(*i));
+               }
+               return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
+       }
+
+       if (options & expand_options::expand_function_args)
+               return abs(arg.expand(options)).hold();
+       else
+               return abs(arg).hold();
+}
+
+static ex abs_expl_derivative(const ex & arg, const symbol & s)
+{
+       ex diff_arg = arg.diff(s);
+       return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
+}
+
 static void abs_print_latex(const ex & arg, const print_context & c)
 {
        c.s << "{|"; arg.print(c); c.s << "|}";
@@ -212,7 +336,7 @@ static void abs_print_csrc_float(const ex & arg, const print_context & c)
 
 static ex abs_conjugate(const ex & arg)
 {
-       return abs(arg);
+       return abs(arg).hold();
 }
 
 static ex abs_real_part(const ex & arg)
@@ -227,14 +351,47 @@ static ex abs_imag_part(const ex& arg)
 
 static ex abs_power(const ex & arg, const ex & exp)
 {
-       if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
-               return power(arg, exp);
-       else
+       if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
+               if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
+                       return pow(arg, exp);
+               else
+                       return pow(arg, exp/2) * pow(arg.conjugate(), exp/2);
+       } else
                return power(abs(arg), exp).hold();
 }
 
+bool abs_info(const ex & arg, unsigned inf)
+{
+       switch (inf) {
+               case info_flags::integer:
+               case info_flags::even:
+               case info_flags::odd:
+               case info_flags::prime:
+                       return arg.info(inf);
+               case info_flags::nonnegint:
+                       return arg.info(info_flags::integer);
+               case info_flags::nonnegative:
+               case info_flags::real:
+                       return true;
+               case info_flags::negative:
+                       return false;
+               case info_flags::positive:
+                       return arg.info(info_flags::positive) || arg.info(info_flags::negative);
+               case info_flags::has_indices: {
+                       if (arg.info(info_flags::has_indices))
+                               return true;
+                       else
+                               return false;
+               }
+       }
+       return false;
+}
+
 REGISTER_FUNCTION(abs, eval_func(abs_eval).
                        evalf_func(abs_evalf).
+                       expand_func(abs_expand).
+                       expl_derivative_func(abs_expl_derivative).
+                       info_func(abs_info).
                        print_func<print_latex>(abs_print_latex).
                        print_func<print_csrc_float>(abs_print_csrc_float).
                        print_func<print_csrc_double>(abs_print_csrc_float).
@@ -295,9 +452,8 @@ static ex step_series(const ex & arg,
            && !(options & series_options::suppress_branchcut))
                throw (std::domain_error("step_series(): on imaginary axis"));
        
-       epvector seq;
-       seq.push_back(expair(step(arg_pt), _ex0));
-       return pseries(rel,seq);
+       epvector seq { expair(step(arg_pt), _ex0) };
+       return pseries(rel, std::move(seq));
 }
 
 static ex step_conjugate(const ex& arg)
@@ -319,8 +475,8 @@ REGISTER_FUNCTION(step, eval_func(step_eval).
                         evalf_func(step_evalf).
                         series_func(step_series).
                         conjugate_func(step_conjugate).
-                                                               real_part_func(step_real_part).
-                                                               imag_part_func(step_imag_part));
+                        real_part_func(step_real_part).
+                        imag_part_func(step_imag_part));
 
 //////////
 // Complex sign
@@ -374,9 +530,8 @@ static ex csgn_series(const ex & arg,
            && !(options & series_options::suppress_branchcut))
                throw (std::domain_error("csgn_series(): on imaginary axis"));
        
-       epvector seq;
-       seq.push_back(expair(csgn(arg_pt), _ex0));
-       return pseries(rel,seq);
+       epvector seq { expair(csgn(arg_pt), _ex0) };
+       return pseries(rel, std::move(seq));
 }
 
 static ex csgn_conjugate(const ex& arg)
@@ -398,7 +553,7 @@ static ex csgn_power(const ex & arg, const ex & exp)
 {
        if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
                if (ex_to<numeric>(exp).is_odd())
-                       return csgn(arg);
+                       return csgn(arg).hold();
                else
                        return power(csgn(arg), _ex2).hold();
        } else
@@ -482,14 +637,13 @@ static ex eta_series(const ex & x, const ex & y,
            (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
            ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
                        throw (std::domain_error("eta_series(): on discontinuity"));
-       epvector seq;
-       seq.push_back(expair(eta(x_pt,y_pt), _ex0));
-       return pseries(rel,seq);
+       epvector seq { expair(eta(x_pt,y_pt), _ex0) };
+       return pseries(rel, std::move(seq));
 }
 
 static ex eta_conjugate(const ex & x, const ex & y)
 {
-       return -eta(x, y);
+       return -eta(x, y).hold();
 }
 
 static ex eta_real_part(const ex & x, const ex & y)
@@ -587,9 +741,8 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                        // substitute the argument's series expansion
                        ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
                        // maybe that was terminating, so add a proper order term
-                       epvector nseq;
-                       nseq.push_back(expair(Order(_ex1), order));
-                       ser += pseries(rel, nseq);
+                       epvector nseq { expair(Order(_ex1), order) };
+                       ser += pseries(rel, std::move(nseq));
                        // reexpanding it will collapse the series again
                        return ser.series(rel, order);
                        // NB: Of course, this still does not allow us to compute anything
@@ -612,9 +765,8 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                        // substitute the argument's series expansion
                        ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
                        // maybe that was terminating, so add a proper order term
-                       epvector nseq;
-                       nseq.push_back(expair(Order(_ex1), order));
-                       ser += pseries(rel, nseq);
+                       epvector nseq { expair(Order(_ex1), order) };
+                       ser += pseries(rel, std::move(nseq));
                        // reexpanding it will collapse the series again
                        return ser.series(rel, order);
                }
@@ -636,18 +788,33 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                                seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
                        // append an order term:
                        seq.push_back(expair(Order(_ex1), replarg.nops()-1));
-                       return pseries(rel, seq);
+                       return pseries(rel, std::move(seq));
                }
        }
        // all other cases should be safe, by now:
        throw do_taylor();  // caught by function::series()
 }
 
+static ex Li2_conjugate(const ex & x)
+{
+       // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
+       // run along the positive real axis beginning at 1.
+       if (x.info(info_flags::negative)) {
+               return Li2(x).hold();
+       }
+       if (is_exactly_a<numeric>(x) &&
+           (!x.imag_part().is_zero() || x < *_num1_p)) {
+               return Li2(x.conjugate());
+       }
+       return conjugate_function(Li2(x)).hold();
+}
+
 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
                        evalf_func(Li2_evalf).
                        derivative_func(Li2_deriv).
                        series_func(Li2_series).
-                       latex_name("\\mbox{Li}_2"));
+                       conjugate_func(Li2_conjugate).
+                       latex_name("\\mathrm{Li}_2"));
 
 //////////
 // trilogarithm
@@ -661,7 +828,7 @@ static ex Li3_eval(const ex & x)
 }
 
 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
-                       latex_name("\\mbox{Li}_3"));
+                       latex_name("\\mathrm{Li}_3"));
 
 //////////
 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
@@ -672,7 +839,7 @@ static ex zetaderiv_eval(const ex & n, const ex & x)
        if (n.info(info_flags::numeric)) {
                // zetaderiv(0,x) -> zeta(x)
                if (n.is_zero())
-                       return zeta(x);
+                       return zeta(x).hold();
        }
        
        return zetaderiv(n, x).hold();
@@ -759,7 +926,7 @@ static ex binomial_sym(const ex & x, const numeric & y)
        if (y.is_integer()) {
                if (y.is_nonneg_integer()) {
                        const unsigned N = y.to_int();
-                       if (N == 0) return _ex0;
+                       if (N == 0) return _ex1;
                        if (N == 1) return x;
                        ex t = x.expand();
                        for (unsigned i = 2; i <= N; ++i)
@@ -783,7 +950,7 @@ static ex binomial_eval(const ex & x, const ex &y)
                return binomial(x, y).hold();
 }
 
-// At the moment the numeric evaluation of a binomail function always
+// At the moment the numeric evaluation of a binomial function always
 // gives a real number, but if this would be implemented using the gamma
 // function, also complex conjugation should be changed (or rather, deleted).
 static ex binomial_conjugate(const ex & x, const ex & y)
@@ -831,11 +998,10 @@ static ex Order_eval(const ex & x)
 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 {
        // Just wrap the function into a pseries object
-       epvector new_seq;
        GINAC_ASSERT(is_a<symbol>(r.lhs()));
        const symbol &s = ex_to<symbol>(r.lhs());
-       new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
-       return pseries(r, new_seq);
+       epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
+       return pseries(r, std::move(new_seq));
 }
 
 static ex Order_conjugate(const ex & x)
@@ -855,11 +1021,15 @@ static ex Order_imag_part(const ex & x)
        return Order(x).hold();
 }
 
-// Differentiation is handled in function::derivative because of its special requirements
+static ex Order_expl_derivative(const ex & arg, const symbol & s)
+{
+       return Order(arg.diff(s));
+}
 
 REGISTER_FUNCTION(Order, eval_func(Order_eval).
                          series_func(Order_series).
                          latex_name("\\mathcal{O}").
+                         expl_derivative_func(Order_expl_derivative).
                          conjugate_func(Order_conjugate).
                          real_part_func(Order_real_part).
                          imag_part_func(Order_imag_part));
@@ -868,13 +1038,31 @@ REGISTER_FUNCTION(Order, eval_func(Order_eval).
 // Solve linear system
 //////////
 
+static void insert_symbols(exset &es, const ex &e)
+{
+       if (is_a<symbol>(e)) {
+               es.insert(e);
+       } else {
+               for (const ex &sube : e) {
+                       insert_symbols(es, sube);
+               }
+       }
+}
+
+static exset symbolset(const ex &e)
+{
+       exset s;
+       insert_symbols(s, e);
+       return s;
+}
+
 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 {
        // solve a system of linear equations
        if (eqns.info(info_flags::relation_equal)) {
                if (!symbols.info(info_flags::symbol))
                        throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
-               const ex sol = lsolve(lst(eqns),lst(symbols));
+               const ex sol = lsolve(lst{eqns}, lst{symbols});
                
                GINAC_ASSERT(sol.nops()==1);
                GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
@@ -883,20 +1071,20 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
        }
        
        // syntax checks
-       if (!eqns.info(info_flags::list)) {
-               throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
+       if (!(eqns.info(info_flags::list) || eqns.info(info_flags::exprseq))) {
+               throw(std::invalid_argument("lsolve(): 1st argument must be a list, a sequence, or an equation"));
        }
        for (size_t i=0; i<eqns.nops(); i++) {
                if (!eqns.op(i).info(info_flags::relation_equal)) {
                        throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
                }
        }
-       if (!symbols.info(info_flags::list)) {
-               throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
+       if (!(symbols.info(info_flags::list) || symbols.info(info_flags::exprseq))) {
+               throw(std::invalid_argument("lsolve(): 2nd argument must be a list, a sequence, or a symbol"));
        }
        for (size_t i=0; i<symbols.nops(); i++) {
                if (!symbols.op(i).info(info_flags::symbol)) {
-                       throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
+                       throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a sequence of symbols"));
                }
        }
        
@@ -907,8 +1095,10 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
        
        for (size_t r=0; r<eqns.nops(); r++) {
                const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+               const exset syms = symbolset(eq);
                ex linpart = eq;
                for (size_t c=0; c<symbols.nops(); c++) {
+                       if (syms.count(symbols.op(c)) == 0) continue;
                        const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
                        linpart -= co*symbols.op(c);
                        sys(r,c) = co;
@@ -918,11 +1108,13 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
        }
        
        // test if system is linear and fill vars matrix
+       const exset sys_syms = symbolset(sys);
+       const exset rhs_syms = symbolset(rhs);
        for (size_t i=0; i<symbols.nops(); i++) {
                vars(i,0) = symbols.op(i);
-               if (sys.has(symbols.op(i)))
+               if (sys_syms.count(symbols.op(i)) != 0)
                        throw(std::logic_error("lsolve: system is not linear"));
-               if (rhs.has(symbols.op(i)))
+               if (rhs_syms.count(symbols.op(i)) != 0)
                        throw(std::logic_error("lsolve: system is not linear"));
        }
        
@@ -932,12 +1124,12 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
        } catch (const std::runtime_error & e) {
                // Probably singular matrix or otherwise overdetermined system:
                // It is consistent to return an empty list
-               return lst();
+               return lst{};
        }
        GINAC_ASSERT(solution.cols()==1);
        GINAC_ASSERT(solution.rows()==symbols.nops());
        
-       // return list of equations of the form lst(var1==sol1,var2==sol2,...)
+       // return list of equations of the form lst{var1==sol1,var2==sol2,...}
        lst sollist;
        for (size_t i=0; i<symbols.nops(); i++)
                sollist.append(symbols.op(i)==solution(i,0));
@@ -992,9 +1184,24 @@ fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
        do {
                xxprev = xx[side];
                fxprev = fx[side];
-               xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
-               fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
-               if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
+               ex dx_ = ff.subs(x == xx[side]).evalf();
+               if (!is_a<numeric>(dx_))
+                       throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
+               xx[side] += ex_to<numeric>(dx_);
+               // Now check if Newton-Raphson method shot out of the interval 
+               bool bad_shot = (side == 0 && xx[0] < xxprev) || 
+                               (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
+               if (!bad_shot) {
+                       // Compute f(x) only if new x is inside the interval.
+                       // The function might be difficult to compute numerically
+                       // or even ill defined outside the interval. Also it's
+                       // a small optimization. 
+                       ex f_x = f.subs(x == xx[side]).evalf();
+                       if (!is_a<numeric>(f_x))
+                               throw std::runtime_error("fsolve(): function does not evaluate numerically");
+                       fx[side] = ex_to<numeric>(f_x);
+               }
+               if (bad_shot) {
                        // Oops, Newton-Raphson method shot out of the interval.
                        // Restore, and try again with the other side instead!
                        xx[side] = xxprev;
@@ -1002,8 +1209,16 @@ fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
                        side = !side;
                        xxprev = xx[side];
                        fxprev = fx[side];
-                       xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
-                       fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
+
+                       ex dx_ = ff.subs(x == xx[side]).evalf();
+                       if (!is_a<numeric>(dx_))
+                               throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
+                       xx[side] += ex_to<numeric>(dx_);
+
+                       ex f_x = f.subs(x==xx[side]).evalf();
+                       if (!is_a<numeric>(f_x))
+                               throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
+                       fx[side] = ex_to<numeric>(f_x);
                }
                if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
                        // Oops, the root isn't bracketed any more.
@@ -1024,10 +1239,13 @@ fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
                        // determined by the secant between the values xx[0] and xx[1].
                        // Don't set the secant_weight to one because that could disturb
                        // the convergence in some corner cases!
-                       static const double secant_weight = 0.984375;  // == 63/64 < 1
+                       constexpr double secant_weight = 0.984375;  // == 63/64 < 1
                        numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
                            + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
-                       numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
+                       ex fxmid_ = f.subs(x == xxmid).evalf();
+                       if (!is_a<numeric>(fxmid_))
+                               throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
+                       numeric fxmid = ex_to<numeric>(fxmid_);
                        if (fxmid.is_zero()) {
                                // Luck strikes...
                                return xxmid;