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  1. S4 :=
  2. Function[{x, w, y1, y2},
  3. Piecewise[{{(11*(y1 + y2))/24,
  4. w == 1/2 + x && w + x == 1/2 && 4*w > 1}, {(y1 + y2)/2,
  5. Element[x, Reals] && w == 0}, {(2*(y1 + y2))/3,
  6. w == 1/2 + x && w + x == 1/2 && 0 < w < 1/4},
  7. {(1/120)*(4*y1 + y2),
  8. 4*w == 1 && x == -(3/4)}, {((-(11 + 20*x))*y1 + y2)/(480*w),
  9. w + x == -(1/2) && w >= 1 + x && 4*w > 1},
  10. {(11/120)*(3*y1 + 2*y2), (w == 1/2 + x && w + x == 0 &&
  11. 4*w > 1) || (4*w == 1 && x == -(1/4))}, {(11/120)*(2*y1 +
  12. 3*y2), (w == x && w + x == 1/2 && 4*w > 1) || (4*w == 1 &&
  13. 4*x == 1)},
  14. {(1/120)*(y1 + 4*y2),
  15. 4*w == 1 && 4*x == 3}, {(2/15)*(3*y1 + 7*y2),
  16. w == x && w + x == 1 &&
  17. 0 < w < 1/4}, {-((7*y1 + 60*x*y1 - 7*y2)/(120*w)),
  18. w + x == 0 && w >= 1 + x},
  19. {(1/30)*(-57*y1 + 92*y2),
  20. w == 1/2 + x && w + x == 1 &&
  21. 0 < w < 1/4}, {(109*y1 - 460*x*y1 + 121*y2)/(480*w),
  22. w + x == 1/2 && w >= 1 + x},
  23. {(1/90)*(-181*y1 + 286*y2),
  24. w == 1/2 + x && w + x == 1 &&
  25. 0 < w < 1/4}, {-(((5 - 4*x)^4*((-5 + 4*x)*y1 - (5 + 4*x)*y2))/
  26. 3840), 4*w == 1 && 3/4 < x < 5/4},
  27. {-(((5 + 4*x)^4*((-5 + 4*x)*y1 - (5 + 4*x)*y2))/3840),
  28. 4*w == 1 && -(5/4) <
  29. x < -(3/4)}, {(1/960)*(-55 +
  30. 8*x*(-5 + 4*x*(15 + 4*x*(-5 + 2*x))))*((-5 + 4*x)*
  31. y1 - (5 + 4*x)*y2),
  32. 4*w == 1 &&
  33. 1/4 < x <
  34. 3/4}, {(1/960)*(-55 +
  35. 8*x*(5 + 4*x*(15 + 4*x*(5 + 2*x))))*((-5 + 4*x)*
  36. y1 - (5 + 4*x)*y2), 4*w == 1 && -(3/4) < x < -(1/4)},
  37. {-(((115 +
  38. 96*x^2*(-5 + 8*x^2))*((-5 + 4*x)*y1 - (5 + 4*x)*y2))/
  39. 1920), 4*w == 1 && -(1/4) < x <
  40. 1/4}, {(x*(-y1 + y2) + w*(y1 + y2))/(2*w),
  41. w >= 1 + x && w + x >= 1},
  42. {((1 + w + x)^4*((-1 + 9*w - x)*y1 + (1 + w + x)*y2))/(15*w),
  43. 1 + w + x >
  44. 0 && ((4*w > 1 && 1/2 + w + x < 0) || (w >= 1 + x &&
  45. 4*w < 1))},
  46. {((1 + w - x)^4*((1 + w - x)*y1 + (-1 + 9*w + x)*y2))/(15*
  47. w), (4*w < 1 && 1 + w > x && w + x > 1) || (4*w > 1 &&
  48. x < 1 + w && 1/2 + w < x)},
  49. {(y1 - 11*y2 + 20*x*y2)/(480*w),
  50. 1/2 + w == x && w + x >= 1 &&
  51. 4*w > 1}, {(7*y1 - 7*y2 + 60*x*y2)/(120*x),
  52. w == x && w + x >= 1},
  53. {(121*y1 + 109*y2 + 460*x*y2)/(480*w),
  54. w == 1/2 + x && w + x >= 1}, {(8/15)*
  55. w*(-5*w*(-1 + x)^2*(y1 - y2) + w^3*(-y1 + y2) -
  56. 5*w^2*(-1 + x)*(y1 + y2) - 5*(-1 + x)^3*(y1 + y2)),
  57. 0 < w <
  58. 1/4 && ((w + x == 1 &&
  59. 1/2 + w <= x) || (Inequality[1/2, Less, w + x, LessEqual,
  60. 1] && 1/2 + w < x))},
  61. {(8/15)*
  62. w*(w^3*(y1 - y2) + 5*w*(1 + x)^2*(y1 - y2) +
  63. 5*w^2*(1 + x)*(y1 + y2) + 5*(1 + x)^3*(y1 + y2)),
  64. w > 0 && w < 1 + x && w + x <= -(1/2)},
  65. {(4/15)*
  66. x*(5*y1 + 5*y2 - 20*x^2*(3*y1 + y2) + 24*x^3*(4*y1 + y2)),
  67. w == x && 4*w < 1 && Inequality[0, Less, w + x, LessEqual, 1/2]},
  68. {(4/15)*
  69. w*(6*w^3*(y1 - y2) + 10*w*x*(-2 + 3*x)*(y1 - y2) +
  70. 10*w^2*(-1 + 3*x)*(y1 + y2) +
  71. 5*(1 + 6*(-1 + x)*x^2)*(y1 + y2)),
  72. 0 < w < 1/4 &&
  73. x <= 1/2 +
  74. w && ((w + x == 1/2 &&
  75. w <= x) || (Inequality[0, Less, w + x, LessEqual, 1/2] &&
  76. w < x))},
  77. {(4/15)*
  78. w*(-10*w*x*(2 + 3*x)*(y1 - y2) + 6*w^3*(-y1 + y2) -
  79. 10*w^2*(1 + 3*x)*(y1 + y2) -
  80. 5*(-1 + 6*x^2*(1 + x))*(y1 + y2)),
  81. 0 < w < 1/4 &&
  82. x <= w && ((w + x == 0 &&
  83. w <= 1/2 + x) || (Inequality[-(1/2), Less, w + x,
  84. LessEqual, 0] && w < 1/2 + x))},
  85. {-(((1 - 2*x)^2*((-71 + 8*x*(32 + x*(-39 + 16*x)))*
  86. y1 + (-29 + 4*x*(21 - 22*x + 8*x^2))*y2))/(60*w)),
  87. w + x > 1/2 && 1/2 + w == x && 4*w < 1},
  88. {-(((1 + 2*x)^2*((-7 + 8*x*(16 + 57*x + 48*x^2))*
  89. y1 + (-13 + 4*x*(3 + 26*x + 24*x^2))*y2))/(60*w)),
  90. w == 1/2 + x && 0 < w < 1/4 && w + x <= 0},
  91. {((55 + 4*x*(-55 + 4*x*(5 + 4*x*(5 + 2*(5 - 12*x)*x))))*
  92. y1 + (55 - 16*x*(5 + x*(25 + 8*x*(5 - 45*x + 48*x^2))))*
  93. y2)/(240*w), w + x == 1/2 && w < 1/2 + x && 4*w > 1},
  94. {((55 + 16*x*(5 + x*(-25 + 8*x*(5 + (5 - 16*x)*x))))*
  95. y1 + (55 + 4*x*(55 + 4*x*(5 - 4*x*(5 + 2*x*(-5 + 4*x)))))*
  96. y2)/(240*w), w == 1/2 + x && 1/2 < w + x < 1},
  97. {((-1 - 64*x^2*(-5 + 2*x*(10 + x*(-15 + 8*x))))*y1 + y2 -
  98. 4*x*(5 + 16*x*(-5 + 2*x*(5 + x*(-5 + 2*x))))*y2)/(120*x),
  99. w == x && 1/2 < w + x < 1},
  100. {-((1/(120*
  101. w))*((224*w^5 + 10*w*(1 + 2*x)^4 - (1 + 2*x)^5 +
  102. 80*w^4*(3 + 10*x) + 80*w^3*(1 + 4*x*(3 + 4*x)) +
  103. 40*w^2*(-5 + 2*x*(-3 + 6*x + 8*x^2)))*y1 +
  104. (96*w^5 + 10*w*(1 + 2*x)^4 + (1 + 2*x)^5 -
  105. 80*w^4*(3 + 2*x) + 80*w^3*(5 + 4*x*(3 + 2*x)) -
  106. 40*w^2*(7 + 6*x*(3 + 2*x)))*y2)),
  107. (w < 1 + x && w + x < 0 && 4*w > 1) || (w > 1/2 + x &&
  108. w + x > -(1/2) && 4*w < 1)},
  109. {-((1/(120*
  110. w))*((96*w^5 + 10*w*(1 - 2*x)^4 +
  111. 80*w^4*(-3 + 2*x) - (-1 + 2*x)^5 +
  112. 80*w^3*(5 + 4*x*(-3 + 2*x)) -
  113. 40*w^2*(7 + 6*x*(-3 + 2*x)))*y1 +
  114. (224*w^5 + 80*w^4*(3 - 10*x) +
  115. 10*w*(1 - 2*x)^4 + (-1 + 2*x)^5 +
  116. 80*w^3*(1 + 4*x*(-3 + 4*x)) -
  117. 40*w^2*(5 + 2*x*(-3 - 6*x + 8*x^2)))*y2)),
  118. (4*w < 1 && 1/2 + w > x && w + x > 1/2) || (4*w > 1 &&
  119. w < x && w + x < 1)},
  120. {-((1/(120*
  121. w))*((7 + 216*w^5 + 280*w^4*(1 + 3*x) +
  122. 400*w^3*x*(2 + 3*x) +
  123. 10*w*(-1 + 2*x^2)*(3 + 8*x + 6*x^2) +
  124. 120*w^2*(-1 + 6*x^2*(1 + x)) +
  125. 2*x*(15 + 20*x - 4*x^3*(5 + 3*x)))*y1 - 7*y2 +
  126.  
  127. 2*(w + x)*(-5*(3 + 4*x) +
  128. 4*(3*w^4 + x^3*(5 + 3*x) + 3*w^2*x*(5 + 6*x) +
  129. w^3*(5 + 12*x) + w*(-5 + 3*x^2*(5 + 4*x))))*y2)), -(1/
  130. 2) < w + x < 0 && w >= 1 + x},
  131. {(1/(120*
  132. w))*((7 + 2*w*(15 + 20*w - 4*w^3*(5 + 3*w)) - 30*x +
  133. 40*w*(-2 + w^2*(4 + 3*w))*x -
  134. 40*(-1 + 6*w^2*(1 + w))*x^2 + 80*w*(2 + 3*w)*x^3 -
  135. 40*(1 + 3*w)*x^4 + 24*x^5)*y1 +
  136. (-7 + 2*w*(15 + 60*w - 4*w^3*(35 + 27*w)) + 30*x +
  137. 40*w*(-2 + w^2*(20 + 21*w))*x -
  138. 40*(1 + 6*w^2*(3 + 5*w))*x^2 + 80*w*(2 + 9*w)*x^3 +
  139. 40*(1 - 3*w)*x^4 - 24*x^5)*y2),
  140. x < 1/2 + w && w < x &&
  141. w + x >=
  142. 1}, {(1/(30*w))*(x*(5 + 40*x^2 + 4*x^4)*(y1 - y2) -
  143. 20*w^5*(y1 + y2) - 5*w*(1 + 4*x^2*(6 + x^2))*(y1 + y2) +
  144. 20*w^4*((4 - 3*x)*y1 + (4 + 3*x)*y2) +
  145.  
  146. 40*w^2*(2*y1 - x*(3 + (-6 + x)*x)*y1 + 2*y2 +
  147. x*(3 + x*(6 + x))*y2) -
  148. 40*w^3*((3 + x*(-4 + 3*x))*y1 + (3 + x*(4 + 3*x))*y2)),
  149. 1/2 < w - x < 1 && 1/2 < w + x < 1},
  150. {(1/(30*
  151. w))*((2 - 2*w^5 + 10*w^4*(1 + x) - 20*w^3*(1 + x)^2 +
  152. 20*w^2*(1 + x)^3 -
  153. 5*w*(-1 + 2*x*(2 + x)*(2 + x*(2 + x))) +
  154. x*(-5 + 2*x*(10 + x*(10 + x*(5 + x)))))*y1 -
  155. (2 + 18*w^5 - 70*w^4*(1 + x) + 100*w^3*(1 + x)^2 -
  156. 60*w^2*(1 + x)^3 +
  157. 5*w*(-1 + 2*x*(2 + x)*(2 + x*(2 + x))) +
  158. x*(-5 + 2*x*(10 + x*(10 + x*(5 + x)))))*y2),
  159. 1/2 + x < w && w < 1 + x && w + x >= 1},
  160. {-((1/(30*
  161. w))*((2 + 18*w^5 + 70*w^4*(-1 + x) + 100*w^3*(-1 + x)^2 +
  162. 60*w^2*(-1 + x)^3 +
  163. 5*w*(-1 + 2*(-2 + x)*x*(2 + (-2 + x)*x)) +
  164. x*(5 - 2*x*(-10 + x*(10 + (-5 + x)*x))))*y1 - 2*y2 +
  165. (w + x)*(-5*(1 + 4*x) +
  166. 2*(w*(-10 + w*(10 + (-5 + w)*w)) +
  167. w*(20 + w*(-15 + 4*w))*x + (10 + 3*w*(-5 + 2*w))*
  168. x^2 + (-5 + 4*w)*x^3 + x^4))*y2)),
  169. 1/2 < w + x < 1 && w >= 1 + x},
  170. {(2*x^5*(-y1 + y2))/(5*w) + 2*w^4*(y1 + y2) +
  171. 2*x^4*(y1 + y2) + (4/3)*
  172. w*(y1 + 3*(-2 + x)*x^2*y1 + y2 - 3*x^2*(2 + x)*y2) + (2/3)*
  173. w^3*((-4 + 9*x)*y1 - (4 + 9*x)*y2) +
  174. (4/3)*w^2*
  175. x*((-4 + 9*x)*y1 + (4 + 9*x)*y2), (x < w && w + x > 0 &&
  176. 4*w < 1) || (w < 1/2 + x && w + x < 1/2 && 4*w > 1)},
  177. {(1/(120*
  178. w))*((1 + 208*w^5 + 80*w^4*(-3 + 11*x) +
  179. 80*w^3*(-1 + 2*x*(-6 + 7*x)) +
  180. 40*w^2*(5 + 2*x*(3 + 2*x*(-3 + 5*x))) +
  181. 2*x*(5 + 4*x*(5 - 2*x*(-5 + (-5 + x)*x))) +
  182. 10*w*(-1 + 8*x*(-1 + x*(-3 + (-4 + x)*x))))*
  183. y1 - (1 + 48*w^5 - 80*w^4*(3 + 5*x) +
  184. 80*w^3*(5 + 2*x*(6 + x)) -
  185. 40*w^2*(7 + 6*x*(3 + 2*x*(1 + x))) +
  186. 2*x*(5 + 4*x*(5 - 2*x*(-5 + (-5 + x)*x))) +
  187. 10*w*(1 + 8*x*(1 + x*(3 - (-4 + x)*x))))*y2),
  188. 0 < w + x < 1/2 && 1/2 < w - x < 1},
  189. {(1/(120*
  190. w))*((-(1 + 48*w^5 + 80*w^4*(-3 + 5*x) +
  191. 80*w^3*(5 + 2*(-6 + x)*x) +
  192. 40*w^2*(-7 + 6*x*(3 + 2*(-1 + x)*x)) -
  193. 10*w*(-1 + 8*x*(1 + x*(-3 + x*(4 + x)))) +
  194. 2*x*(-5 + 4*x*(5 + 2*x*(-5 + x*(5 + x))))))*
  195. y1 + y2 +
  196. 2*(104*w^5 - 40*w^4*(3 + 11*x) +
  197. 40*w^3*(-1 + 2*x*(6 + 7*x)) -
  198. 20*w^2*(-5 + 2*x*(3 + 2*x*(3 + 5*x))) +
  199. 5*w*(-1 + 8*x*(1 + x*(-3 + x*(4 + x)))) +
  200. x*(-5 + 4*x*(5 + 2*x*(-5 + x*(5 + x)))))*y2),
  201. x < w && w < 1/2 + x && 1/2 < w + x < 1},
  202. {(1/(120*
  203. w))*((-7 + 216*w^5 + 400*w^3*x*(-2 + 3*x) +
  204. 280*w^4*(-1 + 3*x) + 120*w^2*(1 + 6*(-1 + x)*x^2) +
  205. 10*w*(3 + 8*x + 4*x^3*(-4 + 3*x)) -
  206. 2*x*(15 + 4*x*(5 + x^2*(-5 + 3*x))))*y1 + 7*y2 +
  207.  
  208. 2*(w + x)*(5*(3 + 4*x) +
  209. 4*(3*w^4 + x^3*(-5 + 3*x) + 3*w^2*x*(-5 + 6*x) +
  210. w^3*(-5 + 12*x) + w*(5 + 3*x^2*(-5 + 4*x))))*y2),
  211. 0 < w + x < 1/2 && w >= 1 + x},
  212. {(1/(120*
  213. w))*((7 + 24*w^5 - 40*w^4*(1 + 3*x) + 80*w^3*x*(2 + 3*x) -
  214. 40*w^2*(-1 + 6*x^2*(1 + x)) +
  215. 10*w*(3 - 8*x + 4*x^3*(4 + 3*x)) -
  216. 2*x*(15 + 4*x*(-5 + x^2*(5 + 3*x))))*y1 +
  217. (-7 + 216*w^5 - 280*w^4*(1 + 3*x) +
  218. 400*w^3*x*(2 + 3*x) - 120*w^2*(-1 + 6*x^2*(1 + x)) +
  219. 10*w*(3 - 8*x + 4*x^3*(4 + 3*x)) +
  220. 2*x*(15 + 4*x*(-5 + x^2*(5 + 3*x))))*y2),
  221. x < w && w < 1/2 + x &&
  222. w + x >= 1}, {(y1 +
  223. 4*x*(5 + 16*x*(5 + 2*x*(5 + x*(5 + 2*x))))*
  224. y1 + (-1 + 64*x^2*(5 + 2*x*(10 + x*(15 + 8*x))))*y2)/(120*
  225. w), w + x == 0 && w < 1 + x && 4*w > 1},
  226. {((55 + 16*x*(5 + x*(-25 + 8*x*(5 + 45*x + 48*x^2))))*
  227. y1 + (55 + 4*x*(55 + 4*x*(5 + 4*x*(-5 + 2*x*(5 + 12*x)))))*
  228. y2)/(240*w), w == 1/2 + x && 0 < w + x < 1/2},
  229. {((55 + 4*x*(-55 + 4*x*(5 + 4*x*(5 + 2*x*(5 + 4*x)))))*
  230. y1 + (55 + 16*x*(-5 + x*(-25 + 8*x*(-5 + x*(5 + 16*x)))))*
  231. y2)/(240*w), w + x == 1/2 && 1/2 + x < w && w < 1 + x}}, 0]]
  232.  
  233. In[241]:= piecewiseFunction = S4[x, w, y1, y2]
  234.  
  235. In[239]:= oneOfTheConditions = w == 1/2 + x && w + x == 1/2 && 4 w > 1
  236.  
  237. Out[239]= w == 1/2 + x && w + x == 1/2 && 4 w > 1
  238.  
  239. In[240]:= Reduce[oneOfTheConditions, {x, w}]
  240.  
  241. Out[240]= x == 0 && w == 1/2
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