Average Error: 43.1 → 21.5
Time: 38.4s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681 \lor \neg \left(i \le 4.039962480132392563803023222135379910469\right):\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{fma}\left(n, \log 1, \left(0.5 \cdot i + 1\right) \cdot i\right)}{i} - n \cdot \frac{\log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.008529206764181104793998144941724603995681 \lor \neg \left(i \le 4.039962480132392563803023222135379910469\right):\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\

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

\end{array}
double f(double i, double n) {
        double r182077 = 100.0;
        double r182078 = 1.0;
        double r182079 = i;
        double r182080 = n;
        double r182081 = r182079 / r182080;
        double r182082 = r182078 + r182081;
        double r182083 = pow(r182082, r182080);
        double r182084 = r182083 - r182078;
        double r182085 = r182084 / r182081;
        double r182086 = r182077 * r182085;
        return r182086;
}
↓
double f(double i, double n) {
        double r182087 = i;
        double r182088 = -0.008529206764181105;
        bool r182089 = r182087 <= r182088;
        double r182090 = 4.039962480132393;
        bool r182091 = r182087 <= r182090;
        double r182092 = !r182091;
        bool r182093 = r182089 || r182092;
        double r182094 = 100.0;
        double r182095 = n;
        double r182096 = r182087 / r182095;
        double r182097 = 1.0;
        double r182098 = r182096 + r182097;
        double r182099 = 2.0;
        double r182100 = r182095 * r182099;
        double r182101 = pow(r182098, r182100);
        double r182102 = r182097 * r182097;
        double r182103 = r182101 - r182102;
        double r182104 = pow(r182098, r182095);
        double r182105 = r182097 + r182104;
        double r182106 = r182103 / r182105;
        double r182107 = r182106 / r182096;
        double r182108 = r182094 * r182107;
        double r182109 = log(r182097);
        double r182110 = 0.5;
        double r182111 = r182110 * r182087;
        double r182112 = r182111 + r182097;
        double r182113 = r182112 * r182087;
        double r182114 = fma(r182095, r182109, r182113);
        double r182115 = r182114 / r182087;
        double r182116 = r182095 * r182115;
        double r182117 = r182087 * r182087;
        double r182118 = r182117 * r182110;
        double r182119 = r182109 * r182118;
        double r182120 = r182119 / r182087;
        double r182121 = r182095 * r182120;
        double r182122 = r182116 - r182121;
        double r182123 = r182094 * r182122;
        double r182124 = r182093 ? r182108 : r182123;
        return r182124;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.1
Target42.9
Herbie21.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -0.008529206764181105 or 4.039962480132393 < i

    1. Initial program 30.1

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

    3. Applied flip--30.1

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

    4. Simplified30.0

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

    5. Simplified30.0

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

    if -0.008529206764181105 < i < 4.039962480132393

    1. Initial program 50.8

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

    2. Taylor expanded around 0 33.6

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

    3. Simplified33.6

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

    5. Applied div-sub33.6

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

    6. Simplified32.9

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

    7. Simplified16.4

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

  4. Final simplification21.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681 \lor \neg \left(i \le 4.039962480132392563803023222135379910469\right):\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{fma}\left(n, \log 1, \left(0.5 \cdot i + 1\right) \cdot i\right)}{i} - n \cdot \frac{\log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))