Average Error: 47.1 → 16.9
Time: 12.0s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.0390445009989891442:\\
\;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 22.1533166385413587:\\
\;\;\;\;\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r152925 = 100.0;
        double r152926 = 1.0;
        double r152927 = i;
        double r152928 = n;
        double r152929 = r152927 / r152928;
        double r152930 = r152926 + r152929;
        double r152931 = pow(r152930, r152928);
        double r152932 = r152931 - r152926;
        double r152933 = r152932 / r152929;
        double r152934 = r152925 * r152933;
        return r152934;
}
↓
double f(double i, double n) {
        double r152935 = i;
        double r152936 = -0.039044500998989144;
        bool r152937 = r152935 <= r152936;
        double r152938 = 100.0;
        double r152939 = r152938 / r152935;
        double r152940 = 1.0;
        double r152941 = n;
        double r152942 = r152935 / r152941;
        double r152943 = r152940 + r152942;
        double r152944 = pow(r152943, r152941);
        double r152945 = r152944 - r152940;
        double r152946 = 1.0;
        double r152947 = r152946 / r152941;
        double r152948 = r152945 / r152947;
        double r152949 = r152939 * r152948;
        double r152950 = 22.15331663854136;
        bool r152951 = r152935 <= r152950;
        double r152952 = 50.0;
        double r152953 = log(r152940);
        double r152954 = r152953 * r152941;
        double r152955 = r152954 / r152935;
        double r152956 = fma(r152938, r152955, r152938);
        double r152957 = r152935 * r152953;
        double r152958 = r152952 * r152957;
        double r152959 = r152956 - r152958;
        double r152960 = fma(r152935, r152952, r152959);
        double r152961 = r152960 * r152941;
        double r152962 = 2.0;
        double r152963 = r152962 * r152941;
        double r152964 = pow(r152943, r152963);
        double r152965 = r152940 * r152940;
        double r152966 = -r152965;
        double r152967 = r152964 + r152966;
        double r152968 = r152944 + r152940;
        double r152969 = r152967 / r152968;
        double r152970 = r152969 / r152942;
        double r152971 = r152938 * r152970;
        double r152972 = r152951 ? r152961 : r152971;
        double r152973 = r152937 ? r152949 : r152972;
        return r152973;
}

Error

Bits error versus i

Bits error versus n

Target

Original47.1
Target46.8
Herbie16.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.039044500998989144

    1. Initial program 27.3

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

    3. Applied div-inv27.3

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

    4. Applied *-un-lft-identity27.3

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

    5. Applied times-frac27.8

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

    6. Applied associate-*r*27.8

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

    7. Simplified27.8

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

    if -0.039044500998989144 < i < 22.15331663854136

    1. Initial program 57.8

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

    2. Taylor expanded around 0 26.8

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

    3. Simplified26.8

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied associate-/r/9.8

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

    6. Applied associate-*r*9.8

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

    7. Taylor expanded around 0 9.8

      \[\leadsto \color{blue}{\left(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    8. Simplified9.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    if 22.15331663854136 < i

    1. Initial program 31.6

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

    3. Applied flip--31.6

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

    4. Simplified31.6

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\mathsf{fma}\left(i, 50, \mathsf{fma}\left(100, \frac{\log 1 \cdot n}{i}, 100\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))))