Compound Interest

Average Error: 43.1 → 21.5

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681 \lor \neg \left(i \le 4.039962480132392563803023222135379910469\right):\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\log 1 \cdot \left(n - \left(i \cdot 0.5\right) \cdot i\right) + i \cdot \left(i \cdot 0.5 + 1\right)}{i} \cdot n\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r390618 = 100.0;
        double r390619 = 1.0;
        double r390620 = i;
        double r390621 = n;
        double r390622 = r390620 / r390621;
        double r390623 = r390619 + r390622;
        double r390624 = pow(r390623, r390621);
        double r390625 = r390624 - r390619;
        double r390626 = r390625 / r390622;
        double r390627 = r390618 * r390626;
        return r390627;
}

↓

double f(double i, double n) {
        double r390628 = i;
        double r390629 = -0.008529206764181105;
        bool r390630 = r390628 <= r390629;
        double r390631 = 4.039962480132393;
        bool r390632 = r390628 <= r390631;
        double r390633 = !r390632;
        bool r390634 = r390630 || r390633;
        double r390635 = 100.0;
        double r390636 = n;
        double r390637 = r390628 / r390636;
        double r390638 = 1.0;
        double r390639 = r390637 + r390638;
        double r390640 = 2.0;
        double r390641 = r390636 * r390640;
        double r390642 = pow(r390639, r390641);
        double r390643 = r390638 * r390638;
        double r390644 = r390642 - r390643;
        double r390645 = pow(r390639, r390636);
        double r390646 = r390638 + r390645;
        double r390647 = r390644 / r390646;
        double r390648 = r390647 / r390637;
        double r390649 = r390635 * r390648;
        double r390650 = log(r390638);
        double r390651 = 0.5;
        double r390652 = r390628 * r390651;
        double r390653 = r390652 * r390628;
        double r390654 = r390636 - r390653;
        double r390655 = r390650 * r390654;
        double r390656 = r390652 + r390638;
        double r390657 = r390628 * r390656;
        double r390658 = r390655 + r390657;
        double r390659 = r390658 / r390628;
        double r390660 = r390659 * r390636;
        double r390661 = r390635 * r390660;
        double r390662 = r390634 ? r390649 : r390661;
        return r390662;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.9
Herbie	21.5

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

Derivation

1. Split input into 2 regimes

2. if i < -0.008529206764181105 or 4.039962480132393 < i

   1. Initial program 30.1

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

   3. Applied flip-- 30.1

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

   4. Simplified 30.0

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

   5. Simplified 30.0

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

   if -0.008529206764181105 < i < 4.039962480132393

   1. Initial program 50.8

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

   2. Taylor expanded around 0 33.6

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

   3. Simplified 33.6

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

   5. Applied associate-/r/ 16.4

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

   6. Simplified 16.4

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

4. Final simplification 21.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681 \lor \neg \left(i \le 4.039962480132392563803023222135379910469\right):\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\log 1 \cdot \left(n - \left(i \cdot 0.5\right) \cdot i\right) + i \cdot \left(i \cdot 0.5 + 1\right)}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))