Average Error: 43.0 → 18.8
Time: 29.4s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{i} \cdot \left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\
\;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\
\;\;\;\;\frac{100}{i} \cdot \left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\

\end{array}
double f(double i, double n) {
        double r111944 = 100.0;
        double r111945 = 1.0;
        double r111946 = i;
        double r111947 = n;
        double r111948 = r111946 / r111947;
        double r111949 = r111945 + r111948;
        double r111950 = pow(r111949, r111947);
        double r111951 = r111950 - r111945;
        double r111952 = r111951 / r111948;
        double r111953 = r111944 * r111952;
        return r111953;
}
↓
double f(double i, double n) {
        double r111954 = i;
        double r111955 = -0.8489635077384908;
        bool r111956 = r111954 <= r111955;
        double r111957 = 100.0;
        double r111958 = n;
        double r111959 = r111954 / r111958;
        double r111960 = pow(r111959, r111958);
        double r111961 = 1.0;
        double r111962 = r111960 - r111961;
        double r111963 = r111957 * r111962;
        double r111964 = r111963 / r111959;
        double r111965 = 0.5476118163719529;
        bool r111966 = r111954 <= r111965;
        double r111967 = r111957 / r111954;
        double r111968 = 0.5;
        double r111969 = 2.0;
        double r111970 = pow(r111954, r111969);
        double r111971 = log(r111961);
        double r111972 = r111971 * r111958;
        double r111973 = fma(r111968, r111970, r111972);
        double r111974 = fma(r111961, r111954, r111973);
        double r111975 = r111970 * r111971;
        double r111976 = r111968 * r111975;
        double r111977 = r111974 - r111976;
        double r111978 = r111977 * r111958;
        double r111979 = r111967 * r111978;
        double r111980 = r111961 + r111959;
        double r111981 = pow(r111980, r111958);
        double r111982 = r111981 - r111961;
        double r111983 = cbrt(r111982);
        double r111984 = r111983 * r111983;
        double r111985 = r111984 / r111954;
        double r111986 = r111957 * r111985;
        double r111987 = 1.0;
        double r111988 = r111987 / r111958;
        double r111989 = r111983 / r111988;
        double r111990 = r111986 * r111989;
        double r111991 = r111966 ? r111979 : r111990;
        double r111992 = r111956 ? r111964 : r111991;
        return r111992;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.0
Target43.5
Herbie18.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.8489635077384908

    1. Initial program 27.4

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

    2. Taylor expanded around inf 64.0

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

    3. Simplified18.1

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

    if -0.8489635077384908 < i < 0.5476118163719529

    1. Initial program 50.8

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

    2. Taylor expanded around 0 34.2

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

    3. Simplified34.2

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

    5. Applied div-inv34.3

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

    6. Applied *-un-lft-identity34.3

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

    7. Applied times-frac15.8

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

    8. Simplified15.8

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

    10. Applied associate-*r*16.2

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

    11. Simplified16.1

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

    if 0.5476118163719529 < i

    1. Initial program 32.4

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

    3. Applied div-inv32.4

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

    4. Applied add-cube-cbrt32.4

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

    5. Applied times-frac32.4

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

    6. Applied associate-*r*32.4

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{i} \cdot \left(\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))