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1 ------------------------------------------------------------------------------
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2 -- --
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3 -- GNAT RUN-TIME COMPONENTS --
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4 -- --
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5 -- ADA.NUMERICS.GENERIC_ELEMENTARY_FUNCTIONS --
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6 -- --
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7 -- S p e c --
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8 -- --
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131
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9 -- Copyright (C) 2012-2018, Free Software Foundation, Inc. --
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111
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10 -- --
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11 -- This specification is derived from the Ada Reference Manual for use with --
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12 -- GNAT. The copyright notice above, and the license provisions that follow --
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13 -- apply solely to the Post aspects that have been added to the spec. --
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14 -- --
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15 -- GNAT is free software; you can redistribute it and/or modify it under --
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16 -- terms of the GNU General Public License as published by the Free Soft- --
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17 -- ware Foundation; either version 3, or (at your option) any later ver- --
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18 -- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
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19 -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
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20 -- or FITNESS FOR A PARTICULAR PURPOSE. --
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21 -- --
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22 -- As a special exception under Section 7 of GPL version 3, you are granted --
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23 -- additional permissions described in the GCC Runtime Library Exception, --
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24 -- version 3.1, as published by the Free Software Foundation. --
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25 -- --
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26 -- You should have received a copy of the GNU General Public License and --
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27 -- a copy of the GCC Runtime Library Exception along with this program; --
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28 -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
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29 -- <http://www.gnu.org/licenses/>. --
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30 -- --
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31 -- GNAT was originally developed by the GNAT team at New York University. --
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32 -- Extensive contributions were provided by Ada Core Technologies Inc. --
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33 -- --
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34 ------------------------------------------------------------------------------
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35
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36 generic
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37 type Float_Type is digits <>;
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38
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39 package Ada.Numerics.Generic_Elementary_Functions with
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40 SPARK_Mode => On
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41 is
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42 pragma Pure;
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43
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44 -- Preconditions in this unit are meant for analysis only, not for run-time
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45 -- checking, so that the expected exceptions are raised when calling
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46 -- Assert. This is enforced by setting the corresponding assertion policy
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47 -- to Ignore. This is done in the generic spec so that it applies to all
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48 -- instances.
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49
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50 pragma Assertion_Policy (Pre => Ignore);
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51
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52 function Sqrt (X : Float_Type'Base) return Float_Type'Base with
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53 Pre => X >= 0.0,
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54 Post => Sqrt'Result >= 0.0
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55 and then (if X = 0.0 then Sqrt'Result = 0.0)
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56 and then (if X = 1.0 then Sqrt'Result = 1.0)
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57
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58 -- Finally if X is positive, the result of Sqrt is positive (because
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59 -- the sqrt of numbers greater than 1 is greater than or equal to 1,
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60 -- and the sqrt of numbers less than 1 is greater than the argument).
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61
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62 -- This property is useful in particular for static analysis. The
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63 -- property that X is positive is not expressed as (X > 0.0), as
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64 -- the value X may be held in registers that have larger range and
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65 -- precision on some architecture (for example, on x86 using x387
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66 -- FPU, as opposed to SSE2). So, it might be possible for X to be
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67 -- 2.0**(-5000) or so, which could cause the number to compare as
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68 -- greater than 0, but Sqrt would still return a zero result.
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69
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70 -- Note: we use the comparison with Succ (0.0) here because this is
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71 -- more amenable to CodePeer analysis than the use of 'Machine.
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72
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73 and then (if X >= Float_Type'Succ (0.0) then Sqrt'Result > 0.0);
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74
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75 function Log (X : Float_Type'Base) return Float_Type'Base with
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76 Pre => X > 0.0,
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77 Post => (if X = 1.0 then Log'Result = 0.0);
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78
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79 function Log (X, Base : Float_Type'Base) return Float_Type'Base with
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80 Pre => X > 0.0 and Base > 0.0 and Base /= 1.0,
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81 Post => (if X = 1.0 then Log'Result = 0.0);
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82
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83 function Exp (X : Float_Type'Base) return Float_Type'Base with
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84 Post => (if X = 0.0 then Exp'Result = 1.0);
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85
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86 function "**" (Left, Right : Float_Type'Base) return Float_Type'Base with
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87 Pre => (if Left = 0.0 then Right > 0.0) and Left >= 0.0,
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88 Post => "**"'Result >= 0.0
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89 and then (if Right = 0.0 then "**"'Result = 1.0)
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90 and then (if Right = 1.0 then "**"'Result = Left)
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91 and then (if Left = 1.0 then "**"'Result = 1.0)
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92 and then (if Left = 0.0 then "**"'Result = 0.0);
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93
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94 function Sin (X : Float_Type'Base) return Float_Type'Base with
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95 Post => Sin'Result in -1.0 .. 1.0
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96 and then (if X = 0.0 then Sin'Result = 0.0);
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97
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98 function Sin (X, Cycle : Float_Type'Base) return Float_Type'Base with
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99 Pre => Cycle > 0.0,
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100 Post => Sin'Result in -1.0 .. 1.0
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101 and then (if X = 0.0 then Sin'Result = 0.0);
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102
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103 function Cos (X : Float_Type'Base) return Float_Type'Base with
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104 Post => Cos'Result in -1.0 .. 1.0
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105 and then (if X = 0.0 then Cos'Result = 1.0);
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106
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107 function Cos (X, Cycle : Float_Type'Base) return Float_Type'Base with
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108 Pre => Cycle > 0.0,
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109 Post => Cos'Result in -1.0 .. 1.0
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110 and then (if X = 0.0 then Cos'Result = 1.0);
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111
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112 function Tan (X : Float_Type'Base) return Float_Type'Base with
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113 Post => (if X = 0.0 then Tan'Result = 0.0);
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114
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115 function Tan (X, Cycle : Float_Type'Base) return Float_Type'Base with
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116 Pre => Cycle > 0.0
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117 and then abs Float_Type'Base'Remainder (X, Cycle) /= 0.25 * Cycle,
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118 Post => (if X = 0.0 then Tan'Result = 0.0);
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119
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120 function Cot (X : Float_Type'Base) return Float_Type'Base with
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121 Pre => X /= 0.0;
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122
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123 function Cot (X, Cycle : Float_Type'Base) return Float_Type'Base with
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124 Pre => Cycle > 0.0
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125 and then X /= 0.0
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126 and then Float_Type'Base'Remainder (X, Cycle) /= 0.0
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127 and then abs Float_Type'Base'Remainder (X, Cycle) = 0.5 * Cycle;
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128
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129 function Arcsin (X : Float_Type'Base) return Float_Type'Base with
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130 Pre => abs X <= 1.0,
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131 Post => (if X = 0.0 then Arcsin'Result = 0.0);
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132
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133 function Arcsin (X, Cycle : Float_Type'Base) return Float_Type'Base with
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134 Pre => Cycle > 0.0 and abs X <= 1.0,
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135 Post => (if X = 0.0 then Arcsin'Result = 0.0);
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136
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137 function Arccos (X : Float_Type'Base) return Float_Type'Base with
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138 Pre => abs X <= 1.0,
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139 Post => (if X = 1.0 then Arccos'Result = 0.0);
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140
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141 function Arccos (X, Cycle : Float_Type'Base) return Float_Type'Base with
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142 Pre => Cycle > 0.0 and abs X <= 1.0,
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143 Post => (if X = 1.0 then Arccos'Result = 0.0);
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144
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145 function Arctan
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146 (Y : Float_Type'Base;
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147 X : Float_Type'Base := 1.0) return Float_Type'Base
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148 with
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149 Pre => X /= 0.0 or Y /= 0.0,
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150 Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
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151
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152 function Arctan
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153 (Y : Float_Type'Base;
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154 X : Float_Type'Base := 1.0;
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155 Cycle : Float_Type'Base) return Float_Type'Base
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156 with
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157 Pre => Cycle > 0.0 and (X /= 0.0 or Y /= 0.0),
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158 Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
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159
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160 function Arccot
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161 (X : Float_Type'Base;
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162 Y : Float_Type'Base := 1.0) return Float_Type'Base
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163 with
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164 Pre => X /= 0.0 or Y /= 0.0,
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165 Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
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166
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167 function Arccot
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168 (X : Float_Type'Base;
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169 Y : Float_Type'Base := 1.0;
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170 Cycle : Float_Type'Base) return Float_Type'Base
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171 with
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172 Pre => Cycle > 0.0 and (X /= 0.0 or Y /= 0.0),
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173 Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
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174
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175 function Sinh (X : Float_Type'Base) return Float_Type'Base with
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176 Post => (if X = 0.0 then Sinh'Result = 0.0);
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177
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178 function Cosh (X : Float_Type'Base) return Float_Type'Base with
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179 Post => Cosh'Result >= 1.0
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180 and then (if X = 0.0 then Cosh'Result = 1.0);
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181
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182 function Tanh (X : Float_Type'Base) return Float_Type'Base with
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183 Post => Tanh'Result in -1.0 .. 1.0
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184 and then (if X = 0.0 then Tanh'Result = 0.0);
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185
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186 function Coth (X : Float_Type'Base) return Float_Type'Base with
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187 Pre => X /= 0.0,
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188 Post => abs Coth'Result >= 1.0;
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189
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190 function Arcsinh (X : Float_Type'Base) return Float_Type'Base with
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191 Post => (if X = 0.0 then Arcsinh'Result = 0.0);
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192
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193 function Arccosh (X : Float_Type'Base) return Float_Type'Base with
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194 Pre => X >= 1.0,
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195 Post => Arccosh'Result >= 0.0
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196 and then (if X = 1.0 then Arccosh'Result = 0.0);
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197
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198 function Arctanh (X : Float_Type'Base) return Float_Type'Base with
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131
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199 Pre => abs X < 1.0,
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200 Post => (if X = 0.0 then Arctanh'Result = 0.0);
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201
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202 function Arccoth (X : Float_Type'Base) return Float_Type'Base with
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131
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203 Pre => abs X > 1.0;
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204
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205 end Ada.Numerics.Generic_Elementary_Functions;
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