Compound Interest

Average Error: 42.3 → 22.6

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.4608487543202562 \cdot 10^{+112}:\\ \;\;\;\;100 \cdot n + \left(\frac{50}{3} \cdot i + 50\right) \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;n \le -0.5086740071675135:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{1}{n}} - n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le 6.641285833915873 \cdot 10^{-252}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + \left(\frac{50}{3} \cdot i + 50\right) \cdot \left(i \cdot n\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r4420563 = 100.0;
        double r4420564 = 1.0;
        double r4420565 = i;
        double r4420566 = n;
        double r4420567 = r4420565 / r4420566;
        double r4420568 = r4420564 + r4420567;
        double r4420569 = pow(r4420568, r4420566);
        double r4420570 = r4420569 - r4420564;
        double r4420571 = r4420570 / r4420567;
        double r4420572 = r4420563 * r4420571;
        return r4420572;
}

↓

double f(double i, double n) {
        double r4420573 = n;
        double r4420574 = -1.4608487543202562e+112;
        bool r4420575 = r4420573 <= r4420574;
        double r4420576 = 100.0;
        double r4420577 = r4420576 * r4420573;
        double r4420578 = 16.666666666666668;
        double r4420579 = i;
        double r4420580 = r4420578 * r4420579;
        double r4420581 = 50.0;
        double r4420582 = r4420580 + r4420581;
        double r4420583 = r4420579 * r4420573;
        double r4420584 = r4420582 * r4420583;
        double r4420585 = r4420577 + r4420584;
        double r4420586 = -0.5086740071675135;
        bool r4420587 = r4420573 <= r4420586;
        double r4420588 = r4420579 / r4420573;
        double r4420589 = 1.0;
        double r4420590 = r4420588 + r4420589;
        double r4420591 = pow(r4420590, r4420573);
        double r4420592 = r4420589 / r4420573;
        double r4420593 = r4420591 / r4420592;
        double r4420594 = r4420593 - r4420573;
        double r4420595 = r4420576 / r4420579;
        double r4420596 = r4420594 * r4420595;
        double r4420597 = 6.641285833915873e-252;
        bool r4420598 = r4420573 <= r4420597;
        double r4420599 = 0.0;
        double r4420600 = r4420598 ? r4420599 : r4420585;
        double r4420601 = r4420587 ? r4420596 : r4420600;
        double r4420602 = r4420575 ? r4420585 : r4420601;
        return r4420602;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.3
Target	42.2
Herbie	22.6

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

Derivation

1. Split input into 3 regimes

2. if n < -1.4608487543202562e+112 or 6.641285833915873e-252 < n

   1. Initial program 54.3

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

   3. Applied div-inv 54.3

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

   4. Applied *-un-lft-identity 54.3

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

   5. Applied times-frac 54.6

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

   6. Applied associate-*r* 54.6

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

   7. Simplified 54.6

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

   8. Taylor expanded around 0 27.4

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{1}{n}}\]

   9. Simplified 27.4

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}\right) + i}}{\frac{1}{n}}\]

   10. Taylor expanded around inf 23.1

       \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

   11. Simplified 23.1

       \[\leadsto \color{blue}{\left(\left(i \cdot i\right) \cdot n\right) \cdot \frac{50}{3} + n \cdot \left(100 + 50 \cdot i\right)}\]

   12. Taylor expanded around -inf 23.1

       \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

   13. Simplified 23.0

       \[\leadsto \color{blue}{100 \cdot n + \left(i \cdot n\right) \cdot \left(\frac{50}{3} \cdot i + 50\right)}\]

   if -1.4608487543202562e+112 < n < -0.5086740071675135

   1. Initial program 35.5

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

   3. Applied div-inv 35.5

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

   4. Applied *-un-lft-identity 35.5

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

   5. Applied times-frac 35.4

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

   6. Applied associate-*r* 35.5

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

   7. Simplified 35.4

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

   9. Applied div-sub 35.4

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

   10. Simplified 35.3

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

   if -0.5086740071675135 < n < 6.641285833915873e-252

   1. Initial program 18.3

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

   3. Applied div-inv 18.3

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

   4. Applied *-un-lft-identity 18.3

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

   5. Applied times-frac 18.5

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

   6. Applied associate-*r* 18.7

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

   7. Simplified 18.7

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

   8. Taylor expanded around 0 15.7

      \[\leadsto \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.4608487543202562 \cdot 10^{+112}:\\ \;\;\;\;100 \cdot n + \left(\frac{50}{3} \cdot i + 50\right) \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;n \le -0.5086740071675135:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{1}{n}} - n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le 6.641285833915873 \cdot 10^{-252}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + \left(\frac{50}{3} \cdot i + 50\right) \cdot \left(i \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))