Compound Interest

Average Error: 42.0 → 16.2

Time: 25.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -2.7755718047646672 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.5457902375689345:\\ \;\;\;\;\mathsf{fma}\left(i, 50 \cdot n, n \cdot \mathsf{fma}\left(i \cdot i, \frac{50}{3}, 100\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(\frac{i}{n} + 1\right)}^{n}, -100\right)}{i}\\ \end{array}\]

C

double f(double i, double n) {
        double r1785058 = 100.0;
        double r1785059 = 1.0;
        double r1785060 = i;
        double r1785061 = n;
        double r1785062 = r1785060 / r1785061;
        double r1785063 = r1785059 + r1785062;
        double r1785064 = pow(r1785063, r1785061);
        double r1785065 = r1785064 - r1785059;
        double r1785066 = r1785065 / r1785062;
        double r1785067 = r1785058 * r1785066;
        return r1785067;
}

↓

double f(double i, double n) {
        double r1785068 = i;
        double r1785069 = -2.7755718047646672e-12;
        bool r1785070 = r1785068 <= r1785069;
        double r1785071 = 100.0;
        double r1785072 = n;
        double r1785073 = r1785068 / r1785072;
        double r1785074 = log1p(r1785073);
        double r1785075 = r1785074 * r1785072;
        double r1785076 = exp(r1785075);
        double r1785077 = -100.0;
        double r1785078 = fma(r1785071, r1785076, r1785077);
        double r1785079 = r1785078 / r1785073;
        double r1785080 = 4.5457902375689345;
        bool r1785081 = r1785068 <= r1785080;
        double r1785082 = 50.0;
        double r1785083 = r1785082 * r1785072;
        double r1785084 = r1785068 * r1785068;
        double r1785085 = 16.666666666666668;
        double r1785086 = fma(r1785084, r1785085, r1785071);
        double r1785087 = r1785072 * r1785086;
        double r1785088 = fma(r1785068, r1785083, r1785087);
        double r1785089 = 1.0;
        double r1785090 = r1785073 + r1785089;
        double r1785091 = pow(r1785090, r1785072);
        double r1785092 = fma(r1785071, r1785091, r1785077);
        double r1785093 = r1785092 / r1785068;
        double r1785094 = r1785072 * r1785093;
        double r1785095 = r1785081 ? r1785088 : r1785094;
        double r1785096 = r1785070 ? r1785079 : r1785095;
        return r1785096;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.0
Target	42.3
Herbie	16.2

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

Derivation

1. Split input into 3 regimes

2. if i < -2.7755718047646672e-12

   1. Initial program 27.8

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

   2. Simplified 27.8

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

   4. Applied add-exp-log 27.8

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

   5. Simplified 6.6

      \[\leadsto \frac{\mathsf{fma}\left(100, e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}, -100\right)}{\frac{i}{n}}\]

   if -2.7755718047646672e-12 < i < 4.5457902375689345

   1. Initial program 50.1

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

   2. Simplified 50.1

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

   3. Taylor expanded around 0 33.1

      \[\leadsto \frac{\color{blue}{100 \cdot i + \left(50 \cdot {i}^{2} + \frac{50}{3} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   4. Simplified 33.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(i \cdot i, 50, \mathsf{fma}\left(i \cdot i, i \cdot \frac{50}{3}, 100 \cdot i\right)\right)}}{\frac{i}{n}}\]

   5. Taylor expanded around inf 16.8

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

   6. Simplified 16.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, n \cdot 50, n \cdot \mathsf{fma}\left(i \cdot i, \frac{50}{3}, 100\right)\right)}\]

   if 4.5457902375689345 < i

   1. Initial program 31.0

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

   2. Simplified 31.0

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

   4. Applied associate-/r/ 31.0

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

4. Final simplification 16.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.7755718047646672 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.5457902375689345:\\ \;\;\;\;\mathsf{fma}\left(i, 50 \cdot n, n \cdot \mathsf{fma}\left(i \cdot i, \frac{50}{3}, 100\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(\frac{i}{n} + 1\right)}^{n}, -100\right)}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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))))