Compound Interest

Average Error: 42.8 → 21.4

Time: 30.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -3.076493361831821548083157134309908748576 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 \cdot 1 + 1 \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{elif}\;i \le 4.681047647742425900350258684689085185658 \cdot 10^{-271}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{fma}\left(n, \log 1, i \cdot \mathsf{fma}\left(i, 0.5, 1\right)\right) - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1}{i}\right)\\ \mathbf{elif}\;i \le 11.91186691865260804945592099102213978767:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \left(\mathsf{fma}\left(n, \log 1, i \cdot \left(i \cdot 0.5 + 1\right)\right) - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}} \cdot 100\\ \end{array}\]

C

double f(double i, double n) {
        double r5886038 = 100.0;
        double r5886039 = 1.0;
        double r5886040 = i;
        double r5886041 = n;
        double r5886042 = r5886040 / r5886041;
        double r5886043 = r5886039 + r5886042;
        double r5886044 = pow(r5886043, r5886041);
        double r5886045 = r5886044 - r5886039;
        double r5886046 = r5886045 / r5886042;
        double r5886047 = r5886038 * r5886046;
        return r5886047;
}

↓

double f(double i, double n) {
        double r5886048 = i;
        double r5886049 = -3.0764933618318215e-11;
        bool r5886050 = r5886048 <= r5886049;
        double r5886051 = 100.0;
        double r5886052 = n;
        double r5886053 = r5886048 / r5886052;
        double r5886054 = 1.0;
        double r5886055 = r5886053 + r5886054;
        double r5886056 = pow(r5886055, r5886052);
        double r5886057 = 3.0;
        double r5886058 = pow(r5886056, r5886057);
        double r5886059 = pow(r5886054, r5886057);
        double r5886060 = r5886058 - r5886059;
        double r5886061 = r5886056 * r5886056;
        double r5886062 = r5886054 * r5886054;
        double r5886063 = r5886054 * r5886056;
        double r5886064 = r5886062 + r5886063;
        double r5886065 = r5886061 + r5886064;
        double r5886066 = r5886053 * r5886065;
        double r5886067 = r5886060 / r5886066;
        double r5886068 = r5886051 * r5886067;
        double r5886069 = 4.681047647742426e-271;
        bool r5886070 = r5886048 <= r5886069;
        double r5886071 = log(r5886054);
        double r5886072 = 0.5;
        double r5886073 = fma(r5886048, r5886072, r5886054);
        double r5886074 = r5886048 * r5886073;
        double r5886075 = fma(r5886052, r5886071, r5886074);
        double r5886076 = r5886048 * r5886048;
        double r5886077 = r5886076 * r5886072;
        double r5886078 = r5886077 * r5886071;
        double r5886079 = r5886075 - r5886078;
        double r5886080 = r5886079 / r5886048;
        double r5886081 = r5886052 * r5886080;
        double r5886082 = r5886051 * r5886081;
        double r5886083 = 11.911866918652608;
        bool r5886084 = r5886048 <= r5886083;
        double r5886085 = r5886051 / r5886048;
        double r5886086 = r5886048 * r5886072;
        double r5886087 = r5886086 + r5886054;
        double r5886088 = r5886048 * r5886087;
        double r5886089 = fma(r5886052, r5886071, r5886088);
        double r5886090 = r5886089 - r5886078;
        double r5886091 = r5886052 * r5886090;
        double r5886092 = r5886085 * r5886091;
        double r5886093 = r5886056 - r5886054;
        double r5886094 = r5886093 / r5886048;
        double r5886095 = 1.0;
        double r5886096 = r5886095 / r5886052;
        double r5886097 = r5886094 / r5886096;
        double r5886098 = r5886097 * r5886051;
        double r5886099 = r5886084 ? r5886092 : r5886098;
        double r5886100 = r5886070 ? r5886082 : r5886099;
        double r5886101 = r5886050 ? r5886068 : r5886100;
        return r5886101;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.8
Target	42.7
Herbie	21.4

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

Derivation

1. Split input into 4 regimes

2. if i < -3.0764933618318215e-11

   1. Initial program 28.6

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

   3. Applied flip3-- 28.6

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

   4. Applied associate-/l/ 28.6

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

   if -3.0764933618318215e-11 < i < 4.681047647742426e-271

   1. Initial program 50.5

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

   2. Taylor expanded around 0 35.0

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

   3. Simplified 35.0

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

   5. Applied div-inv 35.0

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

   6. Applied *-un-lft-identity 35.0

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

   7. Applied times-frac 15.8

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

   8. Simplified 15.8

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

   10. Applied associate-*r* 16.4

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

   11. Simplified 16.3

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

   if 4.681047647742426e-271 < i < 11.911866918652608

   1. Initial program 51.0

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

   2. Taylor expanded around 0 32.8

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

   3. Simplified 32.8

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

   5. Applied div-inv 32.8

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

   6. Applied *-un-lft-identity 32.8

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

   7. Applied times-frac 16.7

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

   8. Simplified 16.7

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

   10. Applied associate-*r* 16.8

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

   11. Simplified 16.7

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

   if 11.911866918652608 < i

   1. Initial program 31.3

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

   3. Applied div-inv 31.3

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

   4. Applied associate-/r* 31.3

      \[\leadsto 100 \cdot \color{blue}{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}{\frac{1}{n}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 21.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.076493361831821548083157134309908748576 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 \cdot 1 + 1 \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{elif}\;i \le 4.681047647742425900350258684689085185658 \cdot 10^{-271}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{fma}\left(n, \log 1, i \cdot \mathsf{fma}\left(i, 0.5, 1\right)\right) - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1}{i}\right)\\ \mathbf{elif}\;i \le 11.91186691865260804945592099102213978767:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \left(\mathsf{fma}\left(n, \log 1, i \cdot \left(i \cdot 0.5 + 1\right)\right) - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}} \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))))