Compound Interest

Average Error: 42.6 → 21.9

Time: 24.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.1051279060415901 \cdot 10^{+112}:\\ \;\;\;\;\left(n \cdot \frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \left(\log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}} \cdot \sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le -1.0933259526011947 \cdot 10^{+64}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le -0.13295556128930017:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le 4.420752440477652 \cdot 10^{-89}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \end{array}\]

C

double f(double i, double n) {
        double r2111051 = 100.0;
        double r2111052 = 1.0;
        double r2111053 = i;
        double r2111054 = n;
        double r2111055 = r2111053 / r2111054;
        double r2111056 = r2111052 + r2111055;
        double r2111057 = pow(r2111056, r2111054);
        double r2111058 = r2111057 - r2111052;
        double r2111059 = r2111058 / r2111055;
        double r2111060 = r2111051 * r2111059;
        return r2111060;
}

↓

double f(double i, double n) {
        double r2111061 = n;
        double r2111062 = -1.1051279060415901e+112;
        bool r2111063 = r2111061 <= r2111062;
        double r2111064 = i;
        double r2111065 = r2111064 * r2111064;
        double r2111066 = 0.5;
        double r2111067 = r2111065 * r2111066;
        double r2111068 = r2111064 * r2111065;
        double r2111069 = 0.16666666666666666;
        double r2111070 = r2111068 * r2111069;
        double r2111071 = exp(r2111070);
        double r2111072 = cbrt(r2111071);
        double r2111073 = r2111072 * r2111072;
        double r2111074 = log(r2111073);
        double r2111075 = log(r2111072);
        double r2111076 = r2111074 + r2111075;
        double r2111077 = r2111067 + r2111076;
        double r2111078 = r2111077 + r2111064;
        double r2111079 = r2111078 / r2111064;
        double r2111080 = r2111061 * r2111079;
        double r2111081 = 100.0;
        double r2111082 = r2111080 * r2111081;
        double r2111083 = -1.0933259526011947e+64;
        bool r2111084 = r2111061 <= r2111083;
        double r2111085 = 1.0;
        double r2111086 = r2111064 / r2111061;
        double r2111087 = r2111085 + r2111086;
        double r2111088 = pow(r2111087, r2111061);
        double r2111089 = r2111088 - r2111085;
        double r2111090 = r2111085 / r2111061;
        double r2111091 = r2111089 / r2111090;
        double r2111092 = r2111081 / r2111064;
        double r2111093 = r2111091 * r2111092;
        double r2111094 = -0.13295556128930017;
        bool r2111095 = r2111061 <= r2111094;
        double r2111096 = sqrt(r2111071);
        double r2111097 = log(r2111096);
        double r2111098 = r2111067 + r2111097;
        double r2111099 = r2111097 + r2111098;
        double r2111100 = r2111099 + r2111064;
        double r2111101 = r2111100 / r2111064;
        double r2111102 = r2111061 * r2111101;
        double r2111103 = r2111102 * r2111081;
        double r2111104 = 4.420752440477652e-89;
        bool r2111105 = r2111061 <= r2111104;
        double r2111106 = 0.0;
        double r2111107 = r2111105 ? r2111106 : r2111103;
        double r2111108 = r2111095 ? r2111103 : r2111107;
        double r2111109 = r2111084 ? r2111093 : r2111108;
        double r2111110 = r2111063 ? r2111082 : r2111109;
        return r2111110;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.6
Target	42.3
Herbie	21.9

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

Derivation

1. Split input into 4 regimes

2. if n < -1.1051279060415901e+112

   1. Initial program 51.0

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

   2. Taylor expanded around 0 48.3

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

   3. Simplified 48.3

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

   5. Applied associate-/r/ 25.2

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

   7. Applied add-log-exp 25.3

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \color{blue}{\log \left(e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}\right)}\right) + i}{i} \cdot n\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 25.3

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

   10. Applied log-prod 25.3

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

   if -1.1051279060415901e+112 < n < -1.0933259526011947e+64

   1. Initial program 35.7

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

   3. Applied div-inv 35.7

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

   4. Applied *-un-lft-identity 35.7

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

   5. Applied times-frac 35.5

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

   6. Applied associate-*r* 35.6

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

   7. Simplified 35.5

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

   if -1.0933259526011947e+64 < n < -0.13295556128930017 or 4.420752440477652e-89 < n

   1. Initial program 54.4

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

   2. Taylor expanded around 0 32.3

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

   3. Simplified 32.3

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

   5. Applied associate-/r/ 17.9

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

   7. Applied add-log-exp 18.1

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \color{blue}{\log \left(e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}\right)}\right) + i}{i} \cdot n\right)\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 18.1

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

   10. Applied log-prod 18.1

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

   11. Applied associate-+r+ 18.1

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

   if -0.13295556128930017 < n < 4.420752440477652e-89

   1. Initial program 28.2

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

   2. Taylor expanded around 0 21.7

      \[\leadsto \color{blue}{0}\]
3. Recombined 4 regimes into one program.

4. Final simplification 21.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.1051279060415901 \cdot 10^{+112}:\\ \;\;\;\;\left(n \cdot \frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \left(\log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}} \cdot \sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le -1.0933259526011947 \cdot 10^{+64}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le -0.13295556128930017:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le 4.420752440477652 \cdot 10^{-89}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))))