Compound Interest

Average Error: 42.8 → 31.4

Time: 28.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.260760569041402 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}} \cdot 100\\ \mathbf{elif}\;n \le -1.901514792923964:\\ \;\;\;\;100 \cdot \frac{\left(i + i \cdot \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) + i \cdot \left(\frac{1}{2} \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 9.072873111493527 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(i + i \cdot \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) + i \cdot \left(\frac{1}{2} \cdot i\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r5764952 = 100.0;
        double r5764953 = 1.0;
        double r5764954 = i;
        double r5764955 = n;
        double r5764956 = r5764954 / r5764955;
        double r5764957 = r5764953 + r5764956;
        double r5764958 = pow(r5764957, r5764955);
        double r5764959 = r5764958 - r5764953;
        double r5764960 = r5764959 / r5764956;
        double r5764961 = r5764952 * r5764960;
        return r5764961;
}

↓

double f(double i, double n) {
        double r5764962 = n;
        double r5764963 = -1.260760569041402e+85;
        bool r5764964 = r5764962 <= r5764963;
        double r5764965 = i;
        double r5764966 = r5764965 / r5764962;
        double r5764967 = 1.0;
        double r5764968 = r5764966 + r5764967;
        double r5764969 = pow(r5764968, r5764962);
        double r5764970 = r5764969 - r5764967;
        double r5764971 = r5764970 / r5764965;
        double r5764972 = r5764967 / r5764962;
        double r5764973 = r5764971 / r5764972;
        double r5764974 = 100.0;
        double r5764975 = r5764973 * r5764974;
        double r5764976 = -1.901514792923964;
        bool r5764977 = r5764962 <= r5764976;
        double r5764978 = 0.16666666666666666;
        double r5764979 = r5764965 * r5764965;
        double r5764980 = r5764978 * r5764979;
        double r5764981 = r5764965 * r5764980;
        double r5764982 = r5764965 + r5764981;
        double r5764983 = 0.5;
        double r5764984 = r5764983 * r5764965;
        double r5764985 = r5764965 * r5764984;
        double r5764986 = r5764982 + r5764985;
        double r5764987 = r5764986 / r5764966;
        double r5764988 = r5764974 * r5764987;
        double r5764989 = 9.072873111493527e-149;
        bool r5764990 = r5764962 <= r5764989;
        double r5764991 = 0.0;
        double r5764992 = r5764990 ? r5764991 : r5764988;
        double r5764993 = r5764977 ? r5764988 : r5764992;
        double r5764994 = r5764964 ? r5764975 : r5764993;
        return r5764994;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.8
Target	42.4
Herbie	31.4

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

Derivation

1. Split input into 3 regimes

2. if n < -1.260760569041402e+85

   1. Initial program 47.8

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

   3. Applied div-inv 47.8

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

   4. Applied associate-/r* 47.4

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

   if -1.260760569041402e+85 < n < -1.901514792923964 or 9.072873111493527e-149 < n

   1. Initial program 53.8

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

   3. Applied add-log-exp 53.8

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

   4. Applied add-log-exp 53.9

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

   5. Applied diff-log 53.9

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

   6. Simplified 53.9

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

   7. Taylor expanded around 0 32.7

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

   8. Simplified 32.7

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

   if -1.901514792923964 < n < 9.072873111493527e-149

   1. Initial program 23.8

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

   2. Taylor expanded around 0 18.3

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

4. Final simplification 31.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.260760569041402 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}} \cdot 100\\ \mathbf{elif}\;n \le -1.901514792923964:\\ \;\;\;\;100 \cdot \frac{\left(i + i \cdot \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) + i \cdot \left(\frac{1}{2} \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 9.072873111493527 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(i + i \cdot \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) + i \cdot \left(\frac{1}{2} \cdot i\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))))