Compound Interest

Average Error: 43.1 → 22.5

Time: 21.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -5.4348495206706917 \cdot 10^{-14}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.10371457373729495 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;i \le 8.555628525961589 \cdot 10^{241}:\\ \;\;\;\;\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot n}{i} \cdot 100\\ \mathbf{elif}\;i \le 3.67423122849791526 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r154050 = 100.0;
        double r154051 = 1.0;
        double r154052 = i;
        double r154053 = n;
        double r154054 = r154052 / r154053;
        double r154055 = r154051 + r154054;
        double r154056 = pow(r154055, r154053);
        double r154057 = r154056 - r154051;
        double r154058 = r154057 / r154054;
        double r154059 = r154050 * r154058;
        return r154059;
}

↓

double f(double i, double n) {
        double r154060 = i;
        double r154061 = -5.434849520670692e-14;
        bool r154062 = r154060 <= r154061;
        double r154063 = 100.0;
        double r154064 = 1.0;
        double r154065 = n;
        double r154066 = r154060 / r154065;
        double r154067 = r154064 + r154066;
        double r154068 = pow(r154067, r154065);
        double r154069 = r154068 - r154064;
        double r154070 = r154063 * r154069;
        double r154071 = r154070 / r154066;
        double r154072 = 1.103714573737295e-38;
        bool r154073 = r154060 <= r154072;
        double r154074 = r154064 * r154060;
        double r154075 = 0.5;
        double r154076 = 2.0;
        double r154077 = pow(r154060, r154076);
        double r154078 = r154075 * r154077;
        double r154079 = log(r154064);
        double r154080 = r154079 * r154065;
        double r154081 = r154078 + r154080;
        double r154082 = r154074 + r154081;
        double r154083 = r154077 * r154079;
        double r154084 = r154075 * r154083;
        double r154085 = r154082 - r154084;
        double r154086 = r154085 * r154065;
        double r154087 = r154086 / r154060;
        double r154088 = r154063 * r154087;
        double r154089 = 8.555628525961589e+241;
        bool r154090 = r154060 <= r154089;
        double r154091 = r154076 * r154065;
        double r154092 = pow(r154067, r154091);
        double r154093 = r154064 * r154064;
        double r154094 = r154092 - r154093;
        double r154095 = r154068 + r154064;
        double r154096 = r154094 / r154095;
        double r154097 = r154096 * r154065;
        double r154098 = r154097 / r154060;
        double r154099 = r154098 * r154063;
        double r154100 = 3.6742312284979153e+289;
        bool r154101 = r154060 <= r154100;
        double r154102 = 1.0;
        double r154103 = r154080 + r154102;
        double r154104 = r154074 + r154103;
        double r154105 = r154104 - r154064;
        double r154106 = r154105 / r154066;
        double r154107 = r154063 * r154106;
        double r154108 = r154101 ? r154107 : r154071;
        double r154109 = r154090 ? r154099 : r154108;
        double r154110 = r154073 ? r154088 : r154109;
        double r154111 = r154062 ? r154071 : r154110;
        return r154111;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	43.0
Herbie	22.5

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

Derivation

1. Split input into 4 regimes

2. if i < -5.434849520670692e-14 or 3.6742312284979153e+289 < 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 associate-*r/ 30.1

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

   if -5.434849520670692e-14 < i < 1.103714573737295e-38

   1. Initial program 50.8

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

   3. Applied div-inv 50.8

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

   4. Applied add-sqr-sqrt 50.8

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

   5. Applied add-sqr-sqrt 50.8

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

   6. Applied difference-of-squares 50.8

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

   7. Applied times-frac 50.5

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

   8. Simplified 50.5

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

   10. Applied *-un-lft-identity 50.5

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

   11. Applied associate-*l* 50.5

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

   12. Simplified 50.4

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

   13. Taylor expanded around 0 15.3

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

   if 1.103714573737295e-38 < i < 8.555628525961589e+241

   1. Initial program 37.1

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

   3. Applied div-inv 37.1

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

   4. Applied add-sqr-sqrt 37.1

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

   5. Applied add-sqr-sqrt 37.1

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

   6. Applied difference-of-squares 37.1

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

   7. Applied times-frac 37.1

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

   8. Simplified 37.1

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

   10. Applied *-un-lft-identity 37.1

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

   11. Applied associate-*l* 37.1

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

   12. Simplified 37.0

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

   14. Applied flip-- 37.0

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

   15. Simplified 37.0

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

   if 8.555628525961589e+241 < i < 3.6742312284979153e+289

   1. Initial program 28.6

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

   2. Taylor expanded around 0 36.4

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

4. Final simplification 22.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -5.4348495206706917 \cdot 10^{-14}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.10371457373729495 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;i \le 8.555628525961589 \cdot 10^{241}:\\ \;\;\;\;\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot n}{i} \cdot 100\\ \mathbf{elif}\;i \le 3.67423122849791526 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))