Average Error: 42.0 → 16.2
Time: 31.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.7755718047646672 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 100, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.5457902375689345:\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot i\right) \cdot n, \frac{50}{3}, n \cdot \left(100 + i \cdot 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(\frac{i}{n} + 1\right)}^{n}, 100, -100\right)}{i}\\ \end{array}\]
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(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 100, -100\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 4.5457902375689345:\\
\;\;\;\;\mathsf{fma}\left(\left(i \cdot i\right) \cdot n, \frac{50}{3}, n \cdot \left(100 + i \cdot 50\right)\right)\\

\mathbf{else}:\\
\;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(\frac{i}{n} + 1\right)}^{n}, 100, -100\right)}{i}\\

\end{array}
double f(double i, double n) {
        double r3741233 = 100.0;
        double r3741234 = 1.0;
        double r3741235 = i;
        double r3741236 = n;
        double r3741237 = r3741235 / r3741236;
        double r3741238 = r3741234 + r3741237;
        double r3741239 = pow(r3741238, r3741236);
        double r3741240 = r3741239 - r3741234;
        double r3741241 = r3741240 / r3741237;
        double r3741242 = r3741233 * r3741241;
        return r3741242;
}
↓
double f(double i, double n) {
        double r3741243 = i;
        double r3741244 = -2.7755718047646672e-12;
        bool r3741245 = r3741243 <= r3741244;
        double r3741246 = n;
        double r3741247 = r3741243 / r3741246;
        double r3741248 = log1p(r3741247);
        double r3741249 = r3741246 * r3741248;
        double r3741250 = exp(r3741249);
        double r3741251 = 100.0;
        double r3741252 = -100.0;
        double r3741253 = fma(r3741250, r3741251, r3741252);
        double r3741254 = r3741253 / r3741247;
        double r3741255 = 4.5457902375689345;
        bool r3741256 = r3741243 <= r3741255;
        double r3741257 = r3741243 * r3741243;
        double r3741258 = r3741257 * r3741246;
        double r3741259 = 16.666666666666668;
        double r3741260 = 50.0;
        double r3741261 = r3741243 * r3741260;
        double r3741262 = r3741251 + r3741261;
        double r3741263 = r3741246 * r3741262;
        double r3741264 = fma(r3741258, r3741259, r3741263);
        double r3741265 = 1.0;
        double r3741266 = r3741247 + r3741265;
        double r3741267 = pow(r3741266, r3741246);
        double r3741268 = fma(r3741267, r3741251, r3741252);
        double r3741269 = r3741268 / r3741243;
        double r3741270 = r3741246 * r3741269;
        double r3741271 = r3741256 ? r3741264 : r3741270;
        double r3741272 = r3741245 ? r3741254 : r3741271;
        return r3741272;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.0
Target42.3
Herbie16.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

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. Simplified27.8

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

    4. Applied add-exp-log27.9

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

    5. Applied pow-exp27.9

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

    6. Simplified6.6

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

    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. Simplified50.1

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

    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. Simplified33.1

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

    5. Taylor expanded around 0 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. Simplified16.8

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

    if 4.5457902375689345 < i

    1. Initial program 31.0

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

    2. Simplified31.0

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

    4. Applied associate-/r/31.0

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

  4. Final simplification16.2

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

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))))