Compound Interest

Average Error: 42.9 → 22.2

Time: 41.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -7.524185979796087207979591237813621340536 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left({\left(\frac{i}{n} + 1\right)}^{n}, {\left(\frac{i}{n} + 1\right)}^{n}, \left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right) \cdot 1\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.943238893158424573925913136918097734451:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 1.540498293983815894197419121524134450797 \cdot 10^{208}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r5974222 = 100.0;
        double r5974223 = 1.0;
        double r5974224 = i;
        double r5974225 = n;
        double r5974226 = r5974224 / r5974225;
        double r5974227 = r5974223 + r5974226;
        double r5974228 = pow(r5974227, r5974225);
        double r5974229 = r5974228 - r5974223;
        double r5974230 = r5974229 / r5974226;
        double r5974231 = r5974222 * r5974230;
        return r5974231;
}

↓

double f(double i, double n) {
        double r5974232 = i;
        double r5974233 = -7.524185979796087e-08;
        bool r5974234 = r5974232 <= r5974233;
        double r5974235 = 100.0;
        double r5974236 = n;
        double r5974237 = r5974232 / r5974236;
        double r5974238 = 1.0;
        double r5974239 = r5974237 + r5974238;
        double r5974240 = pow(r5974239, r5974236);
        double r5974241 = r5974240 * r5974240;
        double r5974242 = r5974241 * r5974240;
        double r5974243 = r5974238 * r5974238;
        double r5974244 = r5974243 * r5974238;
        double r5974245 = r5974242 - r5974244;
        double r5974246 = r5974240 + r5974238;
        double r5974247 = r5974246 * r5974238;
        double r5974248 = fma(r5974240, r5974240, r5974247);
        double r5974249 = r5974245 / r5974248;
        double r5974250 = r5974249 / r5974237;
        double r5974251 = r5974235 * r5974250;
        double r5974252 = 1.9432388931584246;
        bool r5974253 = r5974232 <= r5974252;
        double r5974254 = log(r5974238);
        double r5974255 = r5974232 * r5974232;
        double r5974256 = 0.5;
        double r5974257 = r5974232 * r5974238;
        double r5974258 = fma(r5974255, r5974256, r5974257);
        double r5974259 = fma(r5974236, r5974254, r5974258);
        double r5974260 = r5974255 * r5974254;
        double r5974261 = r5974260 * r5974256;
        double r5974262 = r5974259 - r5974261;
        double r5974263 = r5974262 / r5974232;
        double r5974264 = r5974235 * r5974263;
        double r5974265 = r5974264 * r5974236;
        double r5974266 = 1.540498293983816e+208;
        bool r5974267 = r5974232 <= r5974266;
        double r5974268 = r5974240 / r5974237;
        double r5974269 = r5974238 / r5974237;
        double r5974270 = r5974268 - r5974269;
        double r5974271 = r5974270 * r5974235;
        double r5974272 = 1.0;
        double r5974273 = fma(r5974238, r5974232, r5974272);
        double r5974274 = fma(r5974236, r5974254, r5974273);
        double r5974275 = r5974274 - r5974238;
        double r5974276 = r5974275 / r5974237;
        double r5974277 = r5974235 * r5974276;
        double r5974278 = r5974267 ? r5974271 : r5974277;
        double r5974279 = r5974253 ? r5974265 : r5974278;
        double r5974280 = r5974234 ? r5974251 : r5974279;
        return r5974280;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	42.1
Herbie	22.2

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

Derivation

1. Split input into 4 regimes

2. if i < -7.524185979796087e-08

   1. Initial program 30.2

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

   3. Applied flip3-- 30.2

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

   4. Simplified 30.2

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

   5. Simplified 30.2

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

   if -7.524185979796087e-08 < i < 1.9432388931584246

   1. Initial program 50.4

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

   2. Taylor expanded around 0 34.1

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

   3. Simplified 34.1

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

   5. Applied associate-/r/ 17.0

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

   6. Applied associate-*r* 17.0

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

   if 1.9432388931584246 < i < 1.540498293983816e+208

   1. Initial program 30.7

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

   3. Applied div-sub 30.7

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

   if 1.540498293983816e+208 < i

   1. Initial program 31.8

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

   2. Taylor expanded around 0 34.0

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

   3. Simplified 34.0

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

4. Final simplification 22.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -7.524185979796087207979591237813621340536 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left({\left(\frac{i}{n} + 1\right)}^{n}, {\left(\frac{i}{n} + 1\right)}^{n}, \left({\left(\frac{i}{n} + 1\right)}^{n} + 1\right) \cdot 1\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.943238893158424573925913136918097734451:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(\left(i \cdot i\right) \cdot \log 1\right) \cdot 0.5}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 1.540498293983815894197419121524134450797 \cdot 10^{208}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(1, i, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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))))