A bisection recursive function behave not as expected (R)

Question

I tried to implement a simple bisection algorithm using recursion. I thought it worked but then i wanted to test its efficiency against using while loop.

For some reason, the recursion stops before the condition is met, while the while loop (pun intended) does what i would expect.

When trying to bisect x/2 untill x==0 (the very limit of the computer basically), its tops at value of x = 0.003051758

Code:

# Trying to compare between recursion and while methods of bisection in terms of efficiency
# However the recursion method behaves in an unexpected way

# install.packages("tictoc")

# Mehod 1: recursion
bisect <- function(x,epsilon=0){

  while(x>epsilon){
    cat(x/2,"\n")
    return(bisect(x/2))
  }
  return(x)

} 
# Method 2: while
bisectwhile <- function(x,epsilon=0){
  while(x>epsilon){
    x <- x/2
    print(x)
  }
  return(x)
}

N <- 100
# M1
tictoc::tic()
bisect(N)
tictoc::toc()

    #50 
    #25 
    #.
    #.
    #.
    #0.003051758 
    #[1] 0.003051758

# M2
tictoc::tic()
bisectwhile(N)
tictoc::toc()

Answer 1

A loop is faster than recursion here:

bisect_recursive <- function(x) if(x == 0) x else bisect_recursive(x/2)
bisect_loop <- function(x) { while(x != 0) x <- x/2; 0 }

N <- 100

microbenchmark::microbenchmark(bisect_recursive(N), bisect_loop(N))
#> Unit: microseconds
#>                 expr   min     lq    mean median    uq    max neval cld
#>  bisect_recursive(N) 916.0 941.95 961.515  953.3 962.8 1334.3   100   b
#>       bisect_loop(N) 491.1 499.20 512.297  508.5 516.6  628.1   100  a 

If you want to display all the values, try:

bisect_recursive <- function(x) {
  cat("x is", x, "\n")
  if(x != 0) bisect_recursive(x/2)
}

bisect_loop <- function(x) { 
  while(x != 0) {
    cat("x is", x, "\n")
    x <- x/2 
  }
  cat("x is", x, "\n")
}

Both of which end up at 0, as you can see:

bisect_recursive(100)
#> x is 100 
#> x is 50 
#> x is 25 
#> x is 12.5 
#> x is 6.25 
#> x is 3.125 
#> x is 1.5625 
#> x is 0.78125 
#> x is 0.390625 
#> x is 0.1953125 
#> x is 0.09765625 
#> x is 0.04882812 
#> x is 0.02441406 
#> x is 0.01220703 
#> x is 0.006103516 
#> x is 0.003051758 
#> x is 0.001525879 
#> x is 0.0007629395 
#> x is 0.0003814697 
#> x is 0.0001907349 
#> x is 9.536743e-05 
#> x is 4.768372e-05 
#> x is 2.384186e-05 
#> x is 1.192093e-05 
#> x is 5.960464e-06 
#> x is 2.980232e-06 
#> x is 1.490116e-06 
#> x is 7.450581e-07 
#> x is 3.72529e-07 
#> x is 1.862645e-07 
#> x is 9.313226e-08 
#> x is 4.656613e-08 
#> x is 2.328306e-08 
#> x is 1.164153e-08 
#> x is 5.820766e-09 
#> x is 2.910383e-09 
#> x is 1.455192e-09 
#> x is 7.275958e-10 
#> x is 3.637979e-10 
#> x is 1.818989e-10 
#> x is 9.094947e-11 
#> x is 4.547474e-11 
#> x is 2.273737e-11 
#> x is 1.136868e-11 
#> x is 5.684342e-12 
#> x is 2.842171e-12 
#> x is 1.421085e-12 
#> x is 7.105427e-13 
#> x is 3.552714e-13 
#> x is 1.776357e-13 
#> x is 8.881784e-14 
#> x is 4.440892e-14 
#> x is 2.220446e-14 
#> x is 1.110223e-14 
#> x is 5.551115e-15 
#> x is 2.775558e-15 
#> x is 1.387779e-15 
#> x is 6.938894e-16 
#> x is 3.469447e-16 
#> x is 1.734723e-16 
#> x is 8.673617e-17 
#> x is 4.336809e-17 
#> x is 2.168404e-17 
#> x is 1.084202e-17 
#> x is 5.421011e-18 
#> x is 2.710505e-18 
#> x is 1.355253e-18 
#> x is 6.776264e-19 
#> x is 3.388132e-19 
#> x is 1.694066e-19 
#> x is 8.470329e-20 
#> x is 4.235165e-20 
#> x is 2.117582e-20 
#> x is 1.058791e-20 
#> x is 5.293956e-21 
#> x is 2.646978e-21 
#> x is 1.323489e-21 
#> x is 6.617445e-22 
#> x is 3.308722e-22 
#> x is 1.654361e-22 
#> x is 8.271806e-23 
#> x is 4.135903e-23 
#> x is 2.067952e-23 
#> x is 1.033976e-23 
#> x is 5.169879e-24 
#> x is 2.584939e-24 
#> x is 1.29247e-24 
#> x is 6.462349e-25 
#> x is 3.231174e-25 
#> x is 1.615587e-25 
#> x is 8.077936e-26 
#> x is 4.038968e-26 
#> x is 2.019484e-26 
#> x is 1.009742e-26 
#> x is 5.04871e-27 
#> x is 2.524355e-27 
#> x is 1.262177e-27 
#> x is 6.310887e-28 
#> x is 3.155444e-28 
#> x is 1.577722e-28 
#> x is 7.888609e-29 
#> x is 3.944305e-29 
#> x is 1.972152e-29 
#> x is 9.860761e-30 
#> x is 4.930381e-30 
#> x is 2.46519e-30 
#> x is 1.232595e-30 
#> x is 6.162976e-31 
#> x is 3.081488e-31 
#> x is 1.540744e-31 
#> x is 7.70372e-32 
#> x is 3.85186e-32 
#> x is 1.92593e-32 
#> x is 9.62965e-33 
#> x is 4.814825e-33 
#> x is 2.407412e-33 
#> x is 1.203706e-33 
#> x is 6.018531e-34 
#> x is 3.009266e-34 
#> x is 1.504633e-34 
#> x is 7.523164e-35 
#> x is 3.761582e-35 
#> x is 1.880791e-35 
#> x is 9.403955e-36 
#> x is 4.701977e-36 
#> x is 2.350989e-36 
#> x is 1.175494e-36 
#> x is 5.877472e-37 
#> x is 2.938736e-37 
#> x is 1.469368e-37 
#> x is 7.34684e-38 
#> x is 3.67342e-38 
#> x is 1.83671e-38 
#> x is 9.18355e-39 
#> x is 4.591775e-39 
#> x is 2.295887e-39 
#> x is 1.147944e-39 
#> x is 5.739719e-40 
#> x is 2.869859e-40 
#> x is 1.43493e-40 
#> x is 7.174648e-41 
#> x is 3.587324e-41 
#> x is 1.793662e-41 
#> x is 8.96831e-42 
#> x is 4.484155e-42 
#> x is 2.242078e-42 
#> x is 1.121039e-42 
#> x is 5.605194e-43 
#> x is 2.802597e-43 
#> x is 1.401298e-43 
#> x is 7.006492e-44 
#> x is 3.503246e-44 
#> x is 1.751623e-44 
#> x is 8.758115e-45 
#> x is 4.379058e-45 
#> x is 2.189529e-45 
#> x is 1.094764e-45 
#> x is 5.473822e-46 
#> x is 2.736911e-46 
#> x is 1.368456e-46 
#> x is 6.842278e-47 
#> x is 3.421139e-47 
#> x is 1.710569e-47 
#> x is 8.552847e-48 
#> x is 4.276424e-48 
#> x is 2.138212e-48 
#> x is 1.069106e-48 
#> x is 5.345529e-49 
#> x is 2.672765e-49 
#> x is 1.336382e-49 
#> x is 6.681912e-50 
#> x is 3.340956e-50 
#> x is 1.670478e-50 
#> x is 8.35239e-51 
#> x is 4.176195e-51 
#> x is 2.088097e-51 
#> x is 1.044049e-51 
#> x is 5.220244e-52 
#> x is 2.610122e-52 
#> x is 1.305061e-52 
#> x is 6.525304e-53 
#> x is 3.262652e-53 
#> x is 1.631326e-53 
#> x is 8.156631e-54 
#> x is 4.078315e-54 
#> x is 2.039158e-54 
#> x is 1.019579e-54 
#> x is 5.097894e-55 
#> x is 2.548947e-55 
#> x is 1.274474e-55 
#> x is 6.372368e-56 
#> x is 3.186184e-56 
#> x is 1.593092e-56 
#> x is 7.96546e-57 
#> x is 3.98273e-57 
#> x is 1.991365e-57 
#> x is 9.956824e-58 
#> x is 4.978412e-58 
#> x is 2.489206e-58 
#> x is 1.244603e-58 
#> x is 6.223015e-59 
#> x is 3.111508e-59 
#> x is 1.555754e-59 
#> x is 7.778769e-60 
#> x is 3.889385e-60 
#> x is 1.944692e-60 
#> x is 9.723461e-61 
#> x is 4.861731e-61 
#> x is 2.430865e-61 
#> x is 1.215433e-61 
#> x is 6.077163e-62 
#> x is 3.038582e-62 
#> x is 1.519291e-62 
#> x is 7.596454e-63 
#> x is 3.798227e-63 
#> x is 1.899114e-63 
#> x is 9.495568e-64 
#> x is 4.747784e-64 
#> x is 2.373892e-64 
#> x is 1.186946e-64 
#> x is 5.93473e-65 
#> x is 2.967365e-65 
#> x is 1.483682e-65 
#> x is 7.418412e-66 
#> x is 3.709206e-66 
#> x is 1.854603e-66 
#> x is 9.273015e-67 
#> x is 4.636508e-67 
#> x is 2.318254e-67 
#> x is 1.159127e-67 
#> x is 5.795635e-68 
#> x is 2.897817e-68 
#> x is 1.448909e-68 
#> x is 7.244543e-69 
#> x is 3.622272e-69 
#> x is 1.811136e-69 
#> x is 9.055679e-70 
#> x is 4.52784e-70 
#> x is 2.26392e-70 
#> x is 1.13196e-70 
#> x is 5.659799e-71 
#> x is 2.8299e-71 
#> x is 1.41495e-71 
#> x is 7.074749e-72 
#> x is 3.537375e-72 
#> x is 1.768687e-72 
#> x is 8.843437e-73 
#> x is 4.421718e-73 
#> x is 2.210859e-73 
#> x is 1.10543e-73 
#> x is 5.527148e-74 
#> x is 2.763574e-74 
#> x is 1.381787e-74 
#> x is 6.908935e-75 
#> x is 3.454467e-75 
#> x is 1.727234e-75 
#> x is 8.636169e-76 
#> x is 4.318084e-76 
#> x is 2.159042e-76 
#> x is 1.079521e-76 
#> x is 5.397605e-77 
#> x is 2.698803e-77 
#> x is 1.349401e-77 
#> x is 6.747007e-78 
#> x is 3.373503e-78 
#> x is 1.686752e-78 
#> x is 8.433758e-79 
#> x is 4.216879e-79 
#> x is 2.10844e-79 
#> x is 1.05422e-79 
#> x is 5.271099e-80 
#> x is 2.635549e-80 
#> x is 1.317775e-80 
#> x is 6.588874e-81 
#> x is 3.294437e-81 
#> x is 1.647218e-81 
#> x is 8.236092e-82 
#> x is 4.118046e-82 
#> x is 2.059023e-82 
#> x is 1.029512e-82 
#> x is 5.147558e-83 
#> x is 2.573779e-83 
#> x is 1.286889e-83 
#> x is 6.434447e-84 
#> x is 3.217223e-84 
#> x is 1.608612e-84 
#> x is 8.043059e-85 
#> x is 4.021529e-85 
#> x is 2.010765e-85 
#> x is 1.005382e-85 
#> x is 5.026912e-86 
#> x is 2.513456e-86 
#> x is 1.256728e-86 
#> x is 6.28364e-87 
#> x is 3.14182e-87 
#> x is 1.57091e-87 
#> x is 7.85455e-88 
#> x is 3.927275e-88 
#> x is 1.963637e-88 
#> x is 9.818187e-89 
#> x is 4.909093e-89 
#> x is 2.454547e-89 
#> x is 1.227273e-89 
#> x is 6.136367e-90 
#> x is 3.068183e-90 
#> x is 1.534092e-90 
#> x is 7.670459e-91 
#> x is 3.835229e-91 
#> x is 1.917615e-91 
#> x is 9.588073e-92 
#> x is 4.794037e-92 
#> x is 2.397018e-92 
#> x is 1.198509e-92 
#> x is 5.992546e-93 
#> x is 2.996273e-93 
#> x is 1.498136e-93 
#> x is 7.490682e-94 
#> x is 3.745341e-94 
#> x is 1.872671e-94 
#> x is 9.363353e-95 
#> x is 4.681676e-95 
#> x is 2.340838e-95 
#> x is 1.170419e-95 
#> x is 5.852095e-96 
#> x is 2.926048e-96 
#> x is 1.463024e-96 
#> x is 7.315119e-97 
#> x is 3.65756e-97 
#> x is 1.82878e-97 
#> x is 9.143899e-98 
#> x is 4.57195e-98 
#> x is 2.285975e-98 
#> x is 1.142987e-98 
#> x is 5.714937e-99 
#> x is 2.857468e-99 
#> x is 1.428734e-99 
#> x is 7.143671e-100 
#> x is 3.571836e-100 
#> x is 1.785918e-100 
#> x is 8.929589e-101 
#> x is 4.464794e-101 
#> x is 2.232397e-101 
#> x is 1.116199e-101 
#> x is 5.580993e-102 
#> x is 2.790497e-102 
#> x is 1.395248e-102 
#> x is 6.976241e-103 
#> x is 3.488121e-103 
#> x is 1.74406e-103 
#> x is 8.720302e-104 
#> x is 4.360151e-104 
#> x is 2.180075e-104 
#> x is 1.090038e-104 
#> x is 5.450189e-105 
#> x is 2.725094e-105 
#> x is 1.362547e-105 
#> x is 6.812736e-106 
#> x is 3.406368e-106 
#> x is 1.703184e-106 
#> x is 8.51592e-107 
#> x is 4.25796e-107 
#> x is 2.12898e-107 
#> x is 1.06449e-107 
#> x is 5.32245e-108 
#> x is 2.661225e-108 
#> x is 1.330612e-108 
#> x is 6.653062e-109 
#> x is 3.326531e-109 
#> x is 1.663266e-109 
#> x is 8.316328e-110 
#> x is 4.158164e-110 
#> x is 2.079082e-110 
#> x is 1.039541e-110 
#> x is 5.197705e-111 
#> x is 2.598852e-111 
#> x is 1.299426e-111 
#> x is 6.497131e-112 
#> x is 3.248566e-112 
#> x is 1.624283e-112 
#> x is 8.121414e-113 
#> x is 4.060707e-113 
#> x is 2.030353e-113 
#> x is 1.015177e-113 
#> x is 5.075884e-114 
#> x is 2.537942e-114 
#> x is 1.268971e-114 
#> x is 6.344855e-115 
#> x is 3.172427e-115 
#> x is 1.586214e-115 
#> x is 7.931068e-116 
#> x is 3.965534e-116 
#> x is 1.982767e-116 
#> x is 9.913835e-117 
#> x is 4.956918e-117 
#> x is 2.478459e-117 
#> x is 1.239229e-117 
#> x is 6.196147e-118 
#> x is 3.098074e-118 
#> x is 1.549037e-118 
#> x is 7.745184e-119 
#> x is 3.872592e-119 
#> x is 1.936296e-119 
#> x is 9.68148e-120 
#> x is 4.84074e-120 
#> x is 2.42037e-120 
#> x is 1.210185e-120 
#> x is 6.050925e-121 
#> x is 3.025462e-121 
#> x is 1.512731e-121 
#> x is 7.563656e-122 
#> x is 3.781828e-122 
#> x is 1.890914e-122 
#> x is 9.45457e-123 
#> x is 4.727285e-123 
#> x is 2.363643e-123 
#> x is 1.181821e-123 
#> x is 5.909106e-124 
#> x is 2.954553e-124 
#> x is 1.477277e-124 
#> x is 7.386383e-125 
#> x is 3.693191e-125 
#> x is 1.846596e-125 
#> x is 9.232979e-126 
#> x is 4.616489e-126 
#> x is 2.308245e-126 
#> x is 1.154122e-126 
#> x is 5.770612e-127 
#> x is 2.885306e-127 
#> x is 1.442653e-127 
#> x is 7.213265e-128 
#> x is 3.606632e-128 
#> x is 1.803316e-128 
#> x is 9.016581e-129 
#> x is 4.50829e-129 
#> x is 2.254145e-129 
#> x is 1.127073e-129 
#> x is 5.635363e-130 
#> x is 2.817681e-130 
#> x is 1.408841e-130 
#> x is 7.044204e-131 
#> x is 3.522102e-131 
#> x is 1.761051e-131 
#> x is 8.805255e-132 
#> x is 4.402627e-132 
#> x is 2.201314e-132 
#> x is 1.100657e-132 
#> x is 5.503284e-133 
#> x is 2.751642e-133 
#> x is 1.375821e-133 
#> x is 6.879105e-134 
#> x is 3.439553e-134 
#> x is 1.719776e-134 
#> x is 8.598881e-135 
#> x is 4.299441e-135 
#> x is 2.14972e-135 
#> x is 1.07486e-135 
#> x is 5.374301e-136 
#> x is 2.68715e-136 
#> x is 1.343575e-136 
#> x is 6.717876e-137 
#> x is 3.358938e-137 
#> x is 1.679469e-137 
#> x is 8.397345e-138 
#> x is 4.198673e-138 
#> x is 2.099336e-138 
#> x is 1.049668e-138 
#> x is 5.248341e-139 
#> x is 2.62417e-139 
#> x is 1.312085e-139 
#> x is 6.560426e-140 
#> x is 3.280213e-140 
#> x is 1.640106e-140 
#> x is 8.200532e-141 
#> x is 4.100266e-141 
#> x is 2.050133e-141 
#> x is 1.025067e-141 
#> x is 5.125333e-142 
#> x is 2.562666e-142 
#> x is 1.281333e-142 
#> x is 6.406666e-143 
#> x is 3.203333e-143 
#> x is 1.601666e-143 
#> x is 8.008332e-144 
#> x is 4.004166e-144 
#> x is 2.002083e-144 
#> x is 1.001042e-144 
#> x is 5.005208e-145 
#> x is 2.502604e-145 
#> x is 1.251302e-145 
#> x is 6.25651e-146 
#> x is 3.128255e-146 
#> x is 1.564127e-146 
#> x is 7.820637e-147 
#> x is 3.910319e-147 
#> x is 1.955159e-147 
#> x is 9.775796e-148 
#> x is 4.887898e-148 
#> x is 2.443949e-148 
#> x is 1.221975e-148 
#> x is 6.109873e-149 
#> x is 3.054936e-149 
#> x is 1.527468e-149 
#> x is 7.637341e-150 
#> x is 3.81867e-150 
#> x is 1.909335e-150 
#> x is 9.546676e-151 
#> x is 4.773338e-151 
#> x is 2.386669e-151 
#> x is 1.193335e-151 
#> x is 5.966673e-152 
#> x is 2.983336e-152 
#> x is 1.491668e-152 
#> x is 7.458341e-153 
#> x is 3.72917e-153 
#> x is 1.864585e-153 
#> x is 9.322926e-154 
#> x is 4.661463e-154 
#> x is 2.330731e-154 
#> x is 1.165366e-154 
#> x is 5.826829e-155 
#> x is 2.913414e-155 
#> x is 1.456707e-155 
#> x is 7.283536e-156 
#> x is 3.641768e-156 
#> x is 1.820884e-156 
#> x is 9.10442e-157 
#> x is 4.55221e-157 
#> x is 2.276105e-157 
#> x is 1.138052e-157 
#> x is 5.690262e-158 
#> x is 2.845131e-158 
#> x is 1.422566e-158 
#> x is 7.112828e-159 
#> x is 3.556414e-159 
#> x is 1.778207e-159 
#> x is 8.891035e-160 
#> x is 4.445517e-160 
#> x is 2.222759e-160 
#> x is 1.111379e-160 
#> x is 5.556897e-161 
#> x is 2.778448e-161 
#> x is 1.389224e-161 
#> x is 6.946121e-162 
#> x is 3.473061e-162 
#> x is 1.73653e-162 
#> x is 8.682651e-163 
#> x is 4.341326e-163 
#> x is 2.170663e-163 
#> x is 1.085331e-163 
#> x is 5.426657e-164 
#> x is 2.713329e-164 
#> x is 1.356664e-164 
#> x is 6.783321e-165 
#> x is 3.391661e-165 
#> x is 1.69583e-165 
#> x is 8.479152e-166 
#> x is 4.239576e-166 
#> x is 2.119788e-166 
#> x is 1.059894e-166 
#> x is 5.29947e-167 
#> x is 2.649735e-167 
#> x is 1.324867e-167 
#> x is 6.624337e-168 
#> x is 3.312169e-168 
#> x is 1.656084e-168 
#> x is 8.280422e-169 
#> x is 4.140211e-169 
#> x is 2.070105e-169 
#> x is 1.035053e-169 
#> x is 5.175264e-170 
#> x is 2.587632e-170 
#> x is 1.293816e-170 
#> x is 6.469079e-171 
#> x is 3.23454e-171 
#> x is 1.61727e-171 
#> x is 8.086349e-172 
#> x is 4.043175e-172 
#> x is 2.021587e-172 
#> x is 1.010794e-172 
#> x is 5.053968e-173 
#> x is 2.526984e-173 
#> x is 1.263492e-173 
#> x is 6.31746e-174 
#> x is 3.15873e-174 
#> x is 1.579365e-174 
#> x is 7.896825e-175 
#> x is 3.948413e-175 
#> x is 1.974206e-175 
#> x is 9.871032e-176 
#> x is 4.935516e-176 
#> x is 2.467758e-176 
#> x is 1.233879e-176 
#> x is 6.169395e-177 
#> x is 3.084697e-177 
#> x is 1.542349e-177 
#> x is 7.711744e-178 
#> x is 3.855872e-178 
#> x is 1.927936e-178 
#> x is 9.639679e-179 
#> x is 4.81984e-179 
#> x is 2.40992e-179 
#> x is 1.20496e-179 
#> x is 6.0248e-180 
#> x is 3.0124e-180 
#> x is 1.5062e-180 
#> x is 7.531e-181 
#> x is 3.7655e-181 
#> x is 1.88275e-181 
#> x is 9.413749e-182 
#> x is 4.706875e-182 
#> x is 2.353437e-182 
#> x is 1.176719e-182 
#> x is 5.883593e-183 
#> x is 2.941797e-183 
#> x is 1.470898e-183 
#> x is 7.354492e-184 
#> x is 3.677246e-184 
#> x is 1.838623e-184 
#> x is 9.193115e-185 
#> x is 4.596557e-185 
#> x is 2.298279e-185 
#> x is 1.149139e-185 
#> x is 5.745697e-186 
#> x is 2.872848e-186 
#> x is 1.436424e-186 
#> x is 7.182121e-187 
#> x is 3.59106e-187 
#> x is 1.79553e-187 
#> x is 8.977651e-188 
#> x is 4.488826e-188 
#> x is 2.244413e-188 
#> x is 1.122206e-188 
#> x is 5.611032e-189 
#> x is 2.805516e-189 
#> x is 1.402758e-189 
#> x is 7.01379e-190 
#> x is 3.506895e-190 
#> x is 1.753447e-190 
#> x is 8.767237e-191 
#> x is 4.383619e-191 
#> x is 2.191809e-191 
#> x is 1.095905e-191 
#> x is 5.479523e-192 
#> x is 2.739762e-192 
#> x is 1.369881e-192 
#> x is 6.849404e-193 
#> x is 3.424702e-193 
#> x is 1.712351e-193 
#> x is 8.561755e-194 
#> x is 4.280878e-194 
#> x is 2.140439e-194 
#> x is 1.070219e-194 
#> x is 5.351097e-195 
#> x is 2.675549e-195 
#> x is 1.337774e-195 
#> x is 6.688871e-196 
#> x is 3.344436e-196 
#> x is 1.672218e-196 
#> x is 8.361089e-197 
#> x is 4.180545e-197 
#> x is 2.090272e-197 
#> x is 1.045136e-197 
#> x is 5.225681e-198 
#> x is 2.61284e-198 
#> x is 1.30642e-198 
#> x is 6.532101e-199 
#> x is 3.26605e-199 
#> x is 1.633025e-199 
#> x is 8.165126e-200 
#> x is 4.082563e-200 
#> x is 2.041282e-200 
#> x is 1.020641e-200 
#> x is 5.103204e-201 
#> x is 2.551602e-201 
#> x is 1.275801e-201 
#> x is 6.379005e-202 
#> x is 3.189502e-202 
#> x is 1.594751e-202 
#> x is 7.973756e-203 
#> x is 3.986878e-203 
#> x is 1.993439e-203 
#> x is 9.967195e-204 
#> x is 4.983597e-204 
#> x is 2.491799e-204 
#> x is 1.245899e-204 
#> x is 6.229497e-205 
#> x is 3.114748e-205 
#> x is 1.557374e-205 
#> x is 7.786871e-206 
#> x is 3.893436e-206 
#> x is 1.946718e-206 
#> x is 9.733589e-207 
#> x is 4.866794e-207 
#> x is 2.433397e-207 
#> x is 1.216699e-207 
#> x is 6.083493e-208 
#> x is 3.041747e-208 
#> x is 1.520873e-208 
#> x is 7.604366e-209 
#> x is 3.802183e-209 
#> x is 1.901092e-209 
#> x is 9.505458e-210 
#> x is 4.752729e-210 
#> x is 2.376364e-210 
#> x is 1.188182e-210 
#> x is 5.940911e-211 
#> x is 2.970456e-211 
#> x is 1.485228e-211 
#> x is 7.426139e-212 
#> x is 3.713069e-212 
#> x is 1.856535e-212 
#> x is 9.282674e-213 
#> x is 4.641337e-213 
#> x is 2.320668e-213 
#> x is 1.160334e-213 
#> x is 5.801671e-214 
#> x is 2.900836e-214 
#> x is 1.450418e-214 
#> x is 7.252089e-215 
#> x is 3.626044e-215 
#> x is 1.813022e-215 
#> x is 9.065111e-216 
#> x is 4.532555e-216 
#> x is 2.266278e-216 
#> x is 1.133139e-216 
#> x is 5.665694e-217 
#> x is 2.832847e-217 
#> x is 1.416424e-217 
#> x is 7.082118e-218 
#> x is 3.541059e-218 
#> x is 1.770529e-218 
#> x is 8.852647e-219 
#> x is 4.426324e-219 
#> x is 2.213162e-219 
#> x is 1.106581e-219 
#> x is 5.532905e-220 
#> x is 2.766452e-220 
#> x is 1.383226e-220 
#> x is 6.916131e-221 
#> x is 3.458065e-221 
#> x is 1.729033e-221 
#> x is 8.645164e-222 
#> x is 4.322582e-222 
#> x is 2.161291e-222 
#> x is 1.080645e-222 
#> x is 5.403227e-223 
#> x is 2.701614e-223 
#> x is 1.350807e-223 
#> x is 6.754034e-224 
#> x is 3.377017e-224 
#> x is 1.688509e-224 
#> x is 8.442543e-225 
#> x is 4.221271e-225 
#> x is 2.110636e-225 
#> x is 1.055318e-225 
#> x is 5.276589e-226 
#> x is 2.638295e-226 
#> x is 1.319147e-226 
#> x is 6.595736e-227 
#> x is 3.297868e-227 
#> x is 1.648934e-227 
#> x is 8.24467e-228 
#> x is 4.122335e-228 
#> x is 2.061168e-228 
#> x is 1.030584e-228 
#> x is 5.152919e-229 
#> x is 2.57646e-229 
#> x is 1.28823e-229 
#> x is 6.441149e-230 
#> x is 3.220574e-230 
#> x is 1.610287e-230 
#> x is 8.051436e-231 
#> x is 4.025718e-231 
#> x is 2.012859e-231 
#> x is 1.006429e-231 
#> x is 5.032147e-232 
#> x is 2.516074e-232 
#> x is 1.258037e-232 
#> x is 6.290184e-233 
#> x is 3.145092e-233 
#> x is 1.572546e-233 
#> x is 7.86273e-234 
#> x is 3.931365e-234 
#> x is 1.965683e-234 
#> x is 9.828413e-235 
#> x is 4.914207e-235 
#> x is 2.457103e-235 
#> x is 1.228552e-235 
#> x is 6.142758e-236 
#> x is 3.071379e-236 
#> x is 1.53569e-236 
#> x is 7.678448e-237 
#> x is 3.839224e-237 
#> x is 1.919612e-237 
#> x is 9.59806e-238 
#> x is 4.79903e-238 
#> x is 2.399515e-238 
#> x is 1.199757e-238 
#> x is 5.998787e-239 
#> x is 2.999394e-239 
#> x is 1.499697e-239 
#> x is 7.498484e-240 
#> x is 3.749242e-240 
#> x is 1.874621e-240 
#> x is 9.373105e-241 
#> x is 4.686553e-241 
#> x is 2.343276e-241 
#> x is 1.171638e-241 
#> x is 5.858191e-242 
#> x is 2.929095e-242 
#> x is 1.464548e-242 
#> x is 7.322738e-243 
#> x is 3.661369e-243 
#> x is 1.830685e-243 
#> x is 9.153423e-244 
#> x is 4.576711e-244 
#> x is 2.288356e-244 
#> x is 1.144178e-244 
#> x is 5.720889e-245 
#> x is 2.860445e-245 
#> x is 1.430222e-245 
#> x is 7.151112e-246 
#> x is 3.575556e-246 
#> x is 1.787778e-246 
#> x is 8.93889e-247 
#> x is 4.469445e-247 
#> x is 2.234722e-247 
#> x is 1.117361e-247 
#> x is 5.586806e-248 
#> x is 2.793403e-248 
#> x is 1.396701e-248 
#> x is 6.983507e-249 
#> x is 3.491754e-249 
#> x is 1.745877e-249 
#> x is 8.729384e-250 
#> x is 4.364692e-250 
#> x is 2.182346e-250 
#> x is 1.091173e-250 
#> x is 5.455865e-251 
#> x is 2.727933e-251 
#> x is 1.363966e-251 
#> x is 6.819832e-252 
#> x is 3.409916e-252 
#> x is 1.704958e-252 
#> x is 8.524789e-253 
#> x is 4.262395e-253 
#> x is 2.131197e-253 
#> x is 1.065599e-253 
#> x is 5.327993e-254 
#> x is 2.663997e-254 
#> x is 1.331998e-254 
#> x is 6.659992e-255 
#> x is 3.329996e-255 
#> x is 1.664998e-255 
#> x is 8.32499e-256 
#> x is 4.162495e-256 
#> x is 2.081247e-256 
#> x is 1.040624e-256 
#> x is 5.203119e-257 
#> x is 2.601559e-257 
#> x is 1.30078e-257 
#> x is 6.503898e-258 
#> x is 3.251949e-258 
#> x is 1.625975e-258 
#> x is 8.129873e-259 
#> x is 4.064936e-259 
#> x is 2.032468e-259 
#> x is 1.016234e-259 
#> x is 5.08117e-260 
#> x is 2.540585e-260 
#> x is 1.270293e-260 
#> x is 6.351463e-261 
#> x is 3.175732e-261 
#> x is 1.587866e-261 
#> x is 7.939329e-262 
#> x is 3.969664e-262 
#> x is 1.984832e-262 
#> x is 9.924161e-263 
#> x is 4.962081e-263 
#> x is 2.48104e-263 
#> x is 1.24052e-263 
#> x is 6.202601e-264 
#> x is 3.1013e-264 
#> x is 1.55065e-264 
#> x is 7.753251e-265 
#> x is 3.876625e-265 
#> x is 1.938313e-265 
#> x is 9.691564e-266 
#> x is 4.845782e-266 
#> x is 2.422891e-266 
#> x is 1.211445e-266 
#> x is 6.057227e-267 
#> x is 3.028614e-267 
#> x is 1.514307e-267 
#> x is 7.571534e-268 
#> x is 3.785767e-268 
#> x is 1.892883e-268 
#> x is 9.464417e-269 
#> x is 4.732209e-269 
#> x is 2.366104e-269 
#> x is 1.183052e-269 
#> x is 5.915261e-270 
#> x is 2.95763e-270 
#> x is 1.478815e-270 
#> x is 7.394076e-271 
#> x is 3.697038e-271 
#> x is 1.848519e-271 
#> x is 9.242595e-272 
#> x is 4.621298e-272 
#> x is 2.310649e-272 
#> x is 1.155324e-272 
#> x is 5.776622e-273 
#> x is 2.888311e-273 
#> x is 1.444156e-273 
#> x is 7.220778e-274 
#> x is 3.610389e-274 
#> x is 1.805194e-274 
#> x is 9.025972e-275 
#> x is 4.512986e-275 
#> x is 2.256493e-275 
#> x is 1.128246e-275 
#> x is 5.641232e-276 
#> x is 2.820616e-276 
#> x is 1.410308e-276 
#> x is 7.051541e-277 
#> x is 3.52577e-277 
#> x is 1.762885e-277 
#> x is 8.814426e-278 
#> x is 4.407213e-278 
#> x is 2.203606e-278 
#> x is 1.101803e-278 
#> x is 5.509016e-279 
#> x is 2.754508e-279 
#> x is 1.377254e-279 
#> x is 6.88627e-280 
#> x is 3.443135e-280 
#> x is 1.721568e-280 
#> x is 8.607838e-281 
#> x is 4.303919e-281 
#> x is 2.151959e-281 
#> x is 1.07598e-281 
#> x is 5.379898e-282 
#> x is 2.689949e-282 
#> x is 1.344975e-282 
#> x is 6.724873e-283 
#> x is 3.362437e-283 
#> x is 1.681218e-283 
#> x is 8.406091e-284 
#> x is 4.203046e-284 
#> x is 2.101523e-284 
#> x is 1.050761e-284 
#> x is 5.253807e-285 
#> x is 2.626904e-285 
#> x is 1.313452e-285 
#> x is 6.567259e-286 
#> x is 3.283629e-286 
#> x is 1.641815e-286 
#> x is 8.209074e-287 
#> x is 4.104537e-287 
#> x is 2.052268e-287 
#> x is 1.026134e-287 
#> x is 5.130671e-288 
#> x is 2.565336e-288 
#> x is 1.282668e-288 
#> x is 6.413339e-289 
#> x is 3.206669e-289 
#> x is 1.603335e-289 
#> x is 8.016673e-290 
#> x is 4.008337e-290 
#> x is 2.004168e-290 
#> x is 1.002084e-290 
#> x is 5.010421e-291 
#> x is 2.50521e-291 
#> x is 1.252605e-291 
#> x is 6.263026e-292 
#> x is 3.131513e-292 
#> x is 1.565757e-292 
#> x is 7.828783e-293 
#> x is 3.914391e-293 
#> x is 1.957196e-293 
#> x is 9.785978e-294 
#> x is 4.892989e-294 
#> x is 2.446495e-294 
#> x is 1.223247e-294 
#> x is 6.116236e-295 
#> x is 3.058118e-295 
#> x is 1.529059e-295 
#> x is 7.645296e-296 
#> x is 3.822648e-296 
#> x is 1.911324e-296 
#> x is 9.556619e-297 
#> x is 4.77831e-297 
#> x is 2.389155e-297 
#> x is 1.194577e-297 
#> x is 5.972887e-298 
#> x is 2.986444e-298 
#> x is 1.493222e-298 
#> x is 7.466109e-299 
#> x is 3.733054e-299 
#> x is 1.866527e-299 
#> x is 9.332636e-300 
#> x is 4.666318e-300 
#> x is 2.333159e-300 
#> x is 1.16658e-300 
#> x is 5.832898e-301 
#> x is 2.916449e-301 
#> x is 1.458224e-301 
#> x is 7.291122e-302 
#> x is 3.645561e-302 
#> x is 1.822781e-302 
#> x is 9.113903e-303 
#> x is 4.556951e-303 
#> x is 2.278476e-303 
#> x is 1.139238e-303 
#> x is 5.696189e-304 
#> x is 2.848095e-304 
#> x is 1.424047e-304 
#> x is 7.120236e-305 
#> x is 3.560118e-305 
#> x is 1.780059e-305 
#> x is 8.900295e-306 
#> x is 4.450148e-306 
#> x is 2.225074e-306 
#> x is 1.112537e-306 
#> x is 5.562685e-307 
#> x is 2.781342e-307 
#> x is 1.390671e-307 
#> x is 6.953356e-308 
#> x is 3.476678e-308 
#> x is 1.738339e-308 
#> x is 8.691695e-309 
#> x is 4.345847e-309 
#> x is 2.172924e-309 
#> x is 1.086462e-309 
#> x is 5.432309e-310 
#> x is 2.716155e-310 
#> x is 1.358077e-310 
#> x is 6.790387e-311 
#> x is 3.395193e-311 
#> x is 1.697597e-311 
#> x is 8.487983e-312 
#> x is 4.243992e-312 
#> x is 2.121996e-312 
#> x is 1.060998e-312 
#> x is 5.304989e-313 
#> x is 2.652495e-313 
#> x is 1.326247e-313 
#> x is 6.631237e-314 
#> x is 3.315618e-314 
#> x is 1.657809e-314 
#> x is 8.289046e-315 
#> x is 4.144523e-315 
#> x is 2.072262e-315 
#> x is 1.036131e-315 
#> x is 5.180654e-316 
#> x is 2.590327e-316 
#> x is 1.295163e-316 
#> x is 6.475817e-317 
#> x is 3.237909e-317 
#> x is 1.618954e-317 
#> x is 8.094772e-318 
#> x is 4.047386e-318 
#> x is 2.023693e-318 
#> x is 1.011846e-318 
#> x is 5.059232e-319 
#> x is 2.529616e-319 
#> x is 1.264808e-319 
#> x is 6.32404e-320 
#> x is 3.16202e-320 
#> x is 1.58101e-320 
#> x is 7.90505e-321 
#> x is 3.952525e-321 
#> x is 1.976263e-321 
#> x is 9.881313e-322 
#> x is 4.940656e-322 
#> x is 2.470328e-322 
#> x is 1.235164e-322 
#> x is 5.928788e-323 
#> x is 2.964394e-323 
#> x is 1.482197e-323 
#> x is 9.881313e-324 
#> x is 4.940656e-324 
#> x is 0

^{Created on 2023-03-06 with reprex v2.0.2}