Compound Interest

Average Error: 43.1 → 21.7

Time: 28.5s

Precision: 64

Math

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

↓

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

C

double f(double i, double n) {
        double r6553555 = 100.0;
        double r6553556 = 1.0;
        double r6553557 = i;
        double r6553558 = n;
        double r6553559 = r6553557 / r6553558;
        double r6553560 = r6553556 + r6553559;
        double r6553561 = pow(r6553560, r6553558);
        double r6553562 = r6553561 - r6553556;
        double r6553563 = r6553562 / r6553559;
        double r6553564 = r6553555 * r6553563;
        return r6553564;
}

↓

double f(double i, double n) {
        double r6553565 = i;
        double r6553566 = -0.008529206764181105;
        bool r6553567 = r6553565 <= r6553566;
        double r6553568 = 100.0;
        double r6553569 = r6553568 / r6553565;
        double r6553570 = 1.0;
        double r6553571 = n;
        double r6553572 = r6553565 / r6553571;
        double r6553573 = r6553570 + r6553572;
        double r6553574 = pow(r6553573, r6553571);
        double r6553575 = r6553574 - r6553570;
        double r6553576 = 1.0;
        double r6553577 = r6553576 / r6553571;
        double r6553578 = r6553575 / r6553577;
        double r6553579 = r6553569 * r6553578;
        double r6553580 = 4.039962480132393;
        bool r6553581 = r6553565 <= r6553580;
        double r6553582 = r6553570 * r6553565;
        double r6553583 = 0.5;
        double r6553584 = r6553565 * r6553583;
        double r6553585 = r6553565 * r6553584;
        double r6553586 = log(r6553570);
        double r6553587 = r6553571 - r6553585;
        double r6553588 = r6553586 * r6553587;
        double r6553589 = r6553585 + r6553588;
        double r6553590 = r6553582 + r6553589;
        double r6553591 = r6553590 / r6553565;
        double r6553592 = r6553571 * r6553591;
        double r6553593 = r6553568 * r6553592;
        double r6553594 = r6553581 ? r6553593 : r6553579;
        double r6553595 = r6553567 ? r6553579 : r6553594;
        return r6553595;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.9
Herbie	21.7

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

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 div-inv 30.1

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

   4. Applied *-un-lft-identity 30.1

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

   5. Applied times-frac 30.6

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

   6. Applied associate-*r* 30.6

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

   7. Simplified 30.6

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

   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

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

4. Final simplification 21.7

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

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))))