Average Error: 43.0 → 19.4
Time: 28.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.237131456279114563109935683102053260005 \cdot 10^{-18} \lor \neg \left(i \le 3222196569.804427623748779296875\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{\frac{i}{n}}\right)}^{n}, -1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.237131456279114563109935683102053260005 \cdot 10^{-18} \lor \neg \left(i \le 3222196569.804427623748779296875\right):\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{\frac{i}{n}}\right)}^{n}, -1\right)}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;\frac{100 \cdot \left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\

\end{array}
double f(double i, double n) {
        double r121956 = 100.0;
        double r121957 = 1.0;
        double r121958 = i;
        double r121959 = n;
        double r121960 = r121958 / r121959;
        double r121961 = r121957 + r121960;
        double r121962 = pow(r121961, r121959);
        double r121963 = r121962 - r121957;
        double r121964 = r121963 / r121960;
        double r121965 = r121956 * r121964;
        return r121965;
}
↓
double f(double i, double n) {
        double r121966 = i;
        double r121967 = -1.2371314562791146e-18;
        bool r121968 = r121966 <= r121967;
        double r121969 = 3222196569.8044276;
        bool r121970 = r121966 <= r121969;
        double r121971 = !r121970;
        bool r121972 = r121968 || r121971;
        double r121973 = 100.0;
        double r121974 = 1.0;
        double r121975 = n;
        double r121976 = r121966 / r121975;
        double r121977 = r121974 + r121976;
        double r121978 = cbrt(r121977);
        double r121979 = r121978 * r121978;
        double r121980 = pow(r121979, r121975);
        double r121981 = cbrt(r121976);
        double r121982 = pow(r121981, r121975);
        double r121983 = -r121974;
        double r121984 = fma(r121980, r121982, r121983);
        double r121985 = r121984 / r121976;
        double r121986 = r121973 * r121985;
        double r121987 = 0.5;
        double r121988 = r121966 * r121966;
        double r121989 = log(r121974);
        double r121990 = r121989 * r121975;
        double r121991 = fma(r121987, r121988, r121990);
        double r121992 = fma(r121974, r121966, r121991);
        double r121993 = 2.0;
        double r121994 = pow(r121966, r121993);
        double r121995 = r121994 * r121989;
        double r121996 = r121987 * r121995;
        double r121997 = r121992 - r121996;
        double r121998 = r121997 * r121975;
        double r121999 = r121973 * r121998;
        double r122000 = r121999 / r121966;
        double r122001 = r121972 ? r121986 : r122000;
        return r122001;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.0
Target42.7
Herbie19.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -1.2371314562791146e-18 or 3222196569.8044276 < i

    1. Initial program 30.3

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

    3. Applied add-cube-cbrt30.5

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

    4. Applied unpow-prod-down30.5

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

    5. Applied fma-neg30.5

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

    6. Taylor expanded around inf 59.7

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

    7. Simplified25.0

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

    if -1.2371314562791146e-18 < i < 3222196569.8044276

    1. Initial program 50.8

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

    3. Applied associate-/r/50.4

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

    4. Taylor expanded around 0 16.9

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

    5. Simplified16.9

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

    7. Applied associate-*l/15.7

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

    8. Applied associate-*r/15.9

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification19.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.237131456279114563109935683102053260005 \cdot 10^{-18} \lor \neg \left(i \le 3222196569.804427623748779296875\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{n}, {\left(\sqrt[3]{\frac{i}{n}}\right)}^{n}, -1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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))))