Compound Interest

Average Error: 42.8 → 21.9

Time: 29.9s

Precision: 64

Math

\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

↓

\[\begin{array}{l} \mathbf{if}\;i \le -3.076493361831821548083157134309908748576 \cdot 10^{-11} \lor \neg \left(i \le 7.743283777926976085836940910667181015015\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(0.5 \cdot i + 1\right) + \left(n - i \cdot \left(0.5 \cdot i\right)\right) \cdot \log 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r119060 = 100.0;
        double r119061 = 1.0;
        double r119062 = i;
        double r119063 = n;
        double r119064 = r119062 / r119063;
        double r119065 = r119061 + r119064;
        double r119066 = pow(r119065, r119063);
        double r119067 = r119066 - r119061;
        double r119068 = r119067 / r119064;
        double r119069 = r119060 * r119068;
        return r119069;
}

↓

double f(double i, double n) {
        double r119070 = i;
        double r119071 = -3.0764933618318215e-11;
        bool r119072 = r119070 <= r119071;
        double r119073 = 7.743283777926976;
        bool r119074 = r119070 <= r119073;
        double r119075 = !r119074;
        bool r119076 = r119072 || r119075;
        double r119077 = n;
        double r119078 = 100.0;
        double r119079 = 1.0;
        double r119080 = r119070 / r119077;
        double r119081 = r119079 + r119080;
        double r119082 = pow(r119081, r119077);
        double r119083 = r119082 - r119079;
        double r119084 = r119078 * r119083;
        double r119085 = r119084 / r119070;
        double r119086 = r119077 * r119085;
        double r119087 = 0.5;
        double r119088 = r119087 * r119070;
        double r119089 = r119088 + r119079;
        double r119090 = r119070 * r119089;
        double r119091 = r119070 * r119088;
        double r119092 = r119077 - r119091;
        double r119093 = log(r119079);
        double r119094 = r119092 * r119093;
        double r119095 = r119090 + r119094;
        double r119096 = r119095 / r119070;
        double r119097 = r119077 * r119078;
        double r119098 = r119096 * r119097;
        double r119099 = r119076 ? r119086 : r119098;
        return r119099;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.8
Target	42.7
Herbie	21.9

\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

1. Split input into 2 regimes

2. if i < -3.0764933618318215e-11 or 7.743283777926976 < i

   1. Initial program 29.6

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied associate-/r/ 30.0

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)}\]

   4. Applied associate-*r* 30.0

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n}\]

   5. Simplified 30.0

      \[\leadsto \color{blue}{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right) \cdot 100}{i}} \cdot n\]

   if -3.0764933618318215e-11 < i < 7.743283777926976

   1. Initial program 50.7

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

   2. Taylor expanded around 0 34.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}}{\frac{i}{n}}\]

   3. Simplified 34.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied pow1 34.0

      \[\leadsto 100 \cdot \color{blue}{{\left(\frac{\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{\frac{i}{n}}\right)}^{1}}\]

   6. Applied pow1 34.0

      \[\leadsto \color{blue}{{100}^{1}} \cdot {\left(\frac{\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{\frac{i}{n}}\right)}^{1}\]

   7. Applied pow-prod-down 34.0

      \[\leadsto \color{blue}{{\left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) + \left(n \cdot \log 1 - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{\frac{i}{n}}\right)}^{1}}\]

   8. Simplified 17.0

      \[\leadsto {\color{blue}{\left(\frac{i \cdot \left(i \cdot 0.5 + 1\right) + \log 1 \cdot \left(n - i \cdot \left(i \cdot 0.5\right)\right)}{i} \cdot \left(n \cdot 100\right)\right)}}^{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 21.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.076493361831821548083157134309908748576 \cdot 10^{-11} \lor \neg \left(i \le 7.743283777926976085836940910667181015015\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(0.5 \cdot i + 1\right) + \left(n - i \cdot \left(0.5 \cdot i\right)\right) \cdot \log 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))