Some light rewording and some more details about what is expected for Exercise 9.

93f575d9 · DIAZ Jerome · 4772382b · 93f575d9
Commit 93f575d9 authored 1 year ago by DIAZ Jerome
--- a/1-ProceduralProgramming/6b-hands-on.ipynb
+++ b/1-ProceduralProgramming/6b-hands-on.ipynb
@@ -13,7 +13,7 @@
   "source": [
    "### __EXERCISE 8: Multiplication by an integer__\n",
    "\n",
-    "Write the `Multiply()` function that computes the approximate product of a real by an integer coefficient and returns an integer. \n",
+    "Write the `Multiply()` function that calculates the product of an approximated real by an integer coefficient and returns an integer. \n",
    "\n",
    "*Reminder*: a real is approximated by `numerator / 2^exponent`; your function should rely upon `ComputePowerOf2Approx()` and `TimesPowerOf2()` functions.\n",
    "\n",
@@ -87,7 +87,12 @@
   "source": [
    "### __EXERCISE 9: display sum of two Multiply__\n",
    "\n",
-    "Write a `DisplaySumOfMultiply` function which will write the computation of the sum of two real numbers through their approximation (see expected result below)\n",
+    "Write a `DisplaySumOfMultiply` function which will write the computation of the sum of two approximated real numbers that have each been multiplied by a different integer coefficient.\n",
+    "\n",
+    "The function is expected to compute and display the following:\n",
+    "$$\n",
+    "\\displaystyle r_1 \\cdot c_1 + r_2 \\cdot c_2 \\simeq (a_1 \\cdot 2^{-b_1}) \\cdot c_1 + (a_2 \\cdot 2^{-b_2}) \\cdot c_2\n",
+    "$$\n",
    "\n",
    "This function will take 5 arguments:\n",
    "\n",

 %% Cell type:markdown id: tags:
 # [Getting started in C++](./) - [Procedural programming](./0-main.ipynb) - [Hands-on 3](./7b-hands-on.ipynb)
 %% Cell type:markdown id: tags:
 ### __EXERCISE 8: Multiplication by an integer__
-Write the `Multiply()` function that computes the approximate product of a real by an integer coefficient and returns an integer.
+Write the `Multiply()` function that calculates the product of an approximated real by an integer coefficient and returns an integer.
 *Reminder*: a real is approximated by `numerator / 2^exponent`; your function should rely upon `ComputePowerOf2Approx()` and `TimesPowerOf2()` functions.
 The arguments of `Multiply()` are the maximum number of bits for approximation, the real and the integer coefficient.
 The new main will be:
 %% Cell type:code id: tags:
 ``` C++17
 int main([[maybe_unused]] int argc, [[maybe_unused]] char** argv)
 {
    for (int nbits = 2; nbits <= 8; nbits += 2)
        DisplayPowerOf2Approx(nbits, 0.65) ;
    std::cout << std::endl;
    for (int nbits = 2; nbits <= 8; nbits += 2)
        DisplayPowerOf2Approx(nbits, 0.35) ;
    std::cout << std::endl;
    for (int nbits = 1; nbits <= 8; ++nbits)
    {
        double exact = 0.65 * 3515;
        int rounded = RoundAsInt(exact);
        int approx = Multiply(nbits, 0.65, 3515);
        std::cout << "[With " << nbits << " bits]: 0.65 * 3515 = "
            << rounded << " ~ " << approx << std::endl;
     }
    return EXIT_SUCCESS;
 }
 ```
 %% Cell type:markdown id: tags:
 _Expected result:_
 ```
 [With 2 bits]: 0.65 ~ 0.75 (3 / 2^2)  [error = 15/100]
 [With 4 bits]: 0.65 ~ 0.625 (10 / 2^4)  [error = 4/100]
 [With 6 bits]: 0.65 ~ 0.65625 (42 / 2^6)  [error = 1/100]
 [With 8 bits]: 0.65 ~ 0.648438 (166 / 2^8)  [error = 0/100]
 [With 2 bits]: 0.35 ~ 0.375 (3 / 2^3)  [error = 7/100]
 [With 4 bits]: 0.35 ~ 0.34375 (11 / 2^5)  [error = 2/100]
 [With 6 bits]: 0.35 ~ 0.351562 (45 / 2^7)  [error = 0/100]
 [With 8 bits]: 0.35 ~ 0.349609 (179 / 2^9)  [error = 0/100]
 [With 1 bits]: 0.65 * 3515 = 2285 ~ 1757
 [With 2 bits]: 0.65 * 3515 = 2285 ~ 2636
 [With 3 bits]: 0.65 * 3515 = 2285 ~ 2196
 [With 4 bits]: 0.65 * 3515 = 2285 ~ 2196
 [With 5 bits]: 0.65 * 3515 = 2285 ~ 2306
 [With 6 bits]: 0.65 * 3515 = 2285 ~ 2306
 [With 7 bits]: 0.65 * 3515 = 2285 ~ 2279
 [With 8 bits]: 0.65 * 3515 = 2285 ~ 2279
 ```
 %% Cell type:markdown id: tags:
 ### __EXERCISE 9: display sum of two Multiply__
-Write a `DisplaySumOfMultiply` function which will write the computation of the sum of two real numbers through their approximation (see expected result below)
+Write a `DisplaySumOfMultiply` function which will write the computation of the sum of two approximated real numbers that have each been multiplied by a different integer coefficient.
+The function is expected to compute and display the following:
+$$
+\displaystyle r_1 \cdot c_1 + r_2 \cdot c_2 \simeq (a_1 \cdot 2^{-b_1}) \cdot c_1 + (a_2 \cdot 2^{-b_2}) \cdot c_2
+$$
 This function will take 5 arguments:
 * The number of bits to use.
 * Two real values.
 * Their associated coefficients.
 New main will look like:
 %% Cell type:code id: tags:
 ``` C++17
 int main([[maybe_unused]] int argc, [[maybe_unused]] char** argv)
 {
    for (int nbits = 2; nbits <= 8; nbits += 2)
        DisplayPowerOf2Approx(nbits, 0.65);
    std::cout << std::endl;
    for (int nbits = 2; nbits <= 8; nbits += 2)
        DisplayPowerOf2Approx(nbits, 0.35);
    std::cout << std::endl;
    for (int nbits = 1; nbits <= 8; ++nbits)
        DisplaySumOfMultiply(nbits, 0.65, 3515, 0.35, 4832); // to compute 0.65 * 3515 + 0.35 * 4832
    return EXIT_SUCCESS;
 }
 ```
 %% Cell type:markdown id: tags:
 *Expected result*:
    [With 2 bits]: 0.65 ~ 0.75 (3 / 2^2)  [error = 15/100]
    [With 4 bits]: 0.65 ~ 0.625 (10 / 2^4)  [error = 4/100]
    [With 6 bits]: 0.65 ~ 0.65625 (42 / 2^6)  [error = 1/100]
    [With 8 bits]: 0.65 ~ 0.648438 (166 / 2^8)  [error = 0/100]
    [With 2 bits]: 0.35 ~ 0.375 (3 / 2^3)  [error = 7/100]
    [With 4 bits]: 0.35 ~ 0.34375 (11 / 2^5)  [error = 2/100]
    [With 6 bits]: 0.35 ~ 0.351562 (45 / 2^7)  [error = 0/100]
    [With 8 bits]: 0.35 ~ 0.349609 (179 / 2^9)  [error = 0/100]
    [With 1 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 2965
    [With 2 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4448
    [With 3 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4008
    [With 4 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3857
    [With 5 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3967
    [With 6 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4004
    [With 7 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3977
    [With 8 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3968
 %% Cell type:markdown id: tags:
 ### __EXERCISE 10: print error in `DisplaySumOfMultiply()`__
 Modify slightly the function defined above to add display of the error; we will express it over 1000 (see exercise 7 which was roughly the same!)
 *Expected result*:
 ```
 [With 2 bits]: 0.65 ~ 0.75 (3/2^2)  [error = 15/100]
 [With 4 bits]: 0.65 ~ 0.625 (10/2^4)  [error = 4/100]
 [With 6 bits]: 0.65 ~ 0.65625 (42/2^6)  [error = 1/100]
 [With 8 bits]: 0.65 ~ 0.648438 (166/2^8)  [error = 0/100]
 [With 2 bits]: 0.35 ~ 0.375 (3/2^3)  [error = 7/100]
 [With 4 bits]: 0.35 ~ 0.34375 (11/2^5)  [error = 2/100]
 [With 6 bits]: 0.35 ~ 0.351562 (45/2^7)  [error = 0/100]
 [With 8 bits]: 0.35 ~ 0.349609 (179/2^9)  [error = 0/100]
 [With 1 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 2965  [error = 254/1000]
 [With 2 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4448  [error = 119/1000]
 [With 3 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4008  [error = 8/1000]
 [With 4 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3857  [error = 30/1000]
 [With 5 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3967  [error = 2/1000]
 [With 6 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4004  [error = 7/1000]
 [With 7 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3977  [error = 0/1000]
 [With 8 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3968  [error = 2/1000]
 ```
 %% Cell type:markdown id: tags:
 ### [optional] __EXERCISE 11: write in output file__
 Modify the program so that the `DisplayPowerOf2Approx` and `DisplaySumOfMultiply` functions take an additional argument: the output stream to which the content should be written.
 The following `main()` which writes part of the outputs in a file should work:
 %% Cell type:code id: tags:
 ``` C++17
 int main([[maybe_unused]] int argc, [[maybe_unused]] char** argv)
 {
    std::ofstream out("/tmp/approx_0.65.txt");
    for (int nbits = 2; nbits <= 8; nbits += 2)
        DisplayPowerOf2Approx(out, nbits, 0.65);
    for (int nbits = 2; nbits <= 8; nbits += 2)
        DisplayPowerOf2Approx(std::cout, nbits, 0.35);
    std::cout << std::endl;
    for (int nbits = 1; nbits <= 8; ++nbits)
        DisplaySumOfMultiply(std::cout, nbits, 0.65, 3515, 0.35, 4832); // to compute 0.65 * 3515 + 0.35 * 4832
    return EXIT_SUCCESS;
 }
 ```
 %% Cell type:markdown id: tags:
 *Expected result*:
 ```
 [With 2 bits]: 0.35 ~ 0.375 (3 / 2^3)  [error = 7/100]
 [With 4 bits]: 0.35 ~ 0.34375 (11 / 2^5)  [error = 2/100]
 [With 6 bits]: 0.35 ~ 0.351562 (45 / 2^7)  [error = 0/100]
 [With 8 bits]: 0.35 ~ 0.349609 (179 / 2^9)  [error = 0/100]
 [With 1 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 2965  [error = 254/1000]
 [With 2 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4448  [error = 119/1000]
 [With 3 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4008  [error = 8/1000]
 [With 4 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3857  [error = 30/1000]
 [With 5 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3967  [error = 2/1000]
 [With 6 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 4004  [error = 7/1000]
 [With 7 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3977  [error = 0/1000]
 [With 8 bits]: 0.65 * 3515 + 0.35 * 4832 = 3976 ~ 3968  [error = 2/1000]
 ```
 %% Cell type:markdown id: tags:
 and `cat /tmp/approx_0.65.txt` should return:
 ```
 [With 2 bits]: 0.65 ~ 0.75 (3 / 2^2)  [error = 15/100]
 [With 4 bits]: 0.65 ~ 0.625 (10 / 2^4)  [error = 4/100]
 [With 6 bits]: 0.65 ~ 0.65625 (42 / 2^6)  [error = 1/100]
 [With 8 bits]: 0.65 ~ 0.648438 (166 / 2^8)  [error = 0/100]
 ```
 %% Cell type:markdown id: tags:
 ### [optional] __EXERCISE 12: function pointers__
 Create a `Loop()` function that takes as an argument :
 * The output stream
 * An initial number of bits
 * A final number of bits
 * An increment to be applied to the number of bits
 * A pointer to a function to be executed for each number of bits
 You will need the following intermediate function to be able to use them in `Loop()` (as a specific signature is expected):
 %% Cell type:code id: tags:
 ``` C++17
 // Declarations
 void Display_065(std::ostream& out, int nbits);
 void Display_035(std::ostream& out, int nbits);
 void Display_065_3515_035_4832(std::ostream& out, int nbits);
 // Definitions
 void Display_065(std::ostream& out, int nbits)
 {
    DisplayPowerOf2Approx(out, nbits, 0.65);
 }
 void Display_035(std::ostream& out, int nbits)
 {
    DisplayPowerOf2Approx(out, nbits, 0.35);
 }
 void Display_065_3515_035_4832(std::ostream& out, int nbits)
 {
    DisplaySumOfMultiply(out, nbits, 0.65, 3515, 0.35, 4832);
 }
 // Main
 int main([[maybe_unused]] int argc, [[maybe_unused]] char** argv)
 {
    std::ofstream out("/tmp/approx_0.65.txt");
    Loop(out, 2, 8, 2, Display_065);
    Loop(std::cout, 2, 8, 2, Display_035);
    std::cout << std::endl;
    Loop(std::cout, 1, 8, 1, Display_065_3515_035_4832);
    return EXIT_SUCCESS;
 }
 ```
 %% Cell type:markdown id: tags:
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