Average Error: 43.1 → 32.9
Time: 23.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -6.30319521589689999 \cdot 10^{84}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -5.65874252440070165 \cdot 10^{63}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.02888823886774177 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 5.5209725096373698 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -6.30319521589689999 \cdot 10^{84}:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -5.65874252440070165 \cdot 10^{63}:\\
\;\;\;\;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)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -1.02888823886774177 \cdot 10^{26}:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -934893.903368213796:\\
\;\;\;\;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)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;n \le 5.5209725096373698 \cdot 10^{-155}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;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)}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r112999 = 100.0;
        double r113000 = 1.0;
        double r113001 = i;
        double r113002 = n;
        double r113003 = r113001 / r113002;
        double r113004 = r113000 + r113003;
        double r113005 = pow(r113004, r113002);
        double r113006 = r113005 - r113000;
        double r113007 = r113006 / r113003;
        double r113008 = r112999 * r113007;
        return r113008;
}
↓
double f(double i, double n) {
        double r113009 = n;
        double r113010 = -6.3031952158969e+84;
        bool r113011 = r113009 <= r113010;
        double r113012 = 100.0;
        double r113013 = 1.0;
        double r113014 = i;
        double r113015 = r113014 / r113009;
        double r113016 = r113013 + r113015;
        double r113017 = pow(r113016, r113009);
        double r113018 = r113017 - r113013;
        double r113019 = r113012 * r113018;
        double r113020 = r113019 / r113015;
        double r113021 = -5.658742524400702e+63;
        bool r113022 = r113009 <= r113021;
        double r113023 = 0.5;
        double r113024 = 2.0;
        double r113025 = pow(r113014, r113024);
        double r113026 = log(r113013);
        double r113027 = r113026 * r113009;
        double r113028 = fma(r113023, r113025, r113027);
        double r113029 = fma(r113013, r113014, r113028);
        double r113030 = r113025 * r113026;
        double r113031 = r113023 * r113030;
        double r113032 = r113029 - r113031;
        double r113033 = r113032 / r113015;
        double r113034 = r113012 * r113033;
        double r113035 = -1.0288882388677418e+26;
        bool r113036 = r113009 <= r113035;
        double r113037 = -934893.9033682138;
        bool r113038 = r113009 <= r113037;
        double r113039 = 1.4472641457797862e-291;
        bool r113040 = r113009 <= r113039;
        double r113041 = r113017 / r113015;
        double r113042 = r113013 / r113015;
        double r113043 = r113041 - r113042;
        double r113044 = r113012 * r113043;
        double r113045 = 5.52097250963737e-155;
        bool r113046 = r113009 <= r113045;
        double r113047 = 1.0;
        double r113048 = fma(r113026, r113009, r113047);
        double r113049 = fma(r113013, r113014, r113048);
        double r113050 = r113049 - r113013;
        double r113051 = r113050 / r113015;
        double r113052 = r113012 * r113051;
        double r113053 = r113046 ? r113052 : r113034;
        double r113054 = r113040 ? r113044 : r113053;
        double r113055 = r113038 ? r113034 : r113054;
        double r113056 = r113036 ? r113020 : r113055;
        double r113057 = r113022 ? r113034 : r113056;
        double r113058 = r113011 ? r113020 : r113057;
        return r113058;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.1
Target43.0
Herbie32.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -6.3031952158969e+84 or -5.658742524400702e+63 < n < -1.0288882388677418e+26

    1. Initial program 45.6

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

    3. Applied associate-*r/45.6

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

    if -6.3031952158969e+84 < n < -5.658742524400702e+63 or -1.0288882388677418e+26 < n < -934893.9033682138 or 5.52097250963737e-155 < n

    1. Initial program 57.0

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

    2. Taylor expanded around 0 34.0

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

    3. Simplified34.0

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

    if -934893.9033682138 < n < 1.4472641457797862e-291

    1. Initial program 18.0

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

    3. Applied div-sub18.0

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

    if 1.4472641457797862e-291 < n < 5.52097250963737e-155

    1. Initial program 43.0

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

    2. Taylor expanded around 0 32.4

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

    3. Simplified32.4

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification32.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.30319521589689999 \cdot 10^{84}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -5.65874252440070165 \cdot 10^{63}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.02888823886774177 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 5.5209725096373698 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))