Average Error: 42.4 → 22.4
Time: 16.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.71552055838372837 \cdot 10^{82}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -3.9974032830534776 \cdot 10^{-251}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\ \mathbf{elif}\;n \le 7.18594828335009282 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.71552055838372837 \cdot 10^{82}:\\
\;\;\;\;100 \cdot \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)\\

\mathbf{elif}\;n \le -3.9974032830534776 \cdot 10^{-251}:\\
\;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\

\mathbf{elif}\;n \le 7.18594828335009282 \cdot 10^{-152}:\\
\;\;\;\;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 \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)\\

\end{array}
double f(double i, double n) {
        double r147125 = 100.0;
        double r147126 = 1.0;
        double r147127 = i;
        double r147128 = n;
        double r147129 = r147127 / r147128;
        double r147130 = r147126 + r147129;
        double r147131 = pow(r147130, r147128);
        double r147132 = r147131 - r147126;
        double r147133 = r147132 / r147129;
        double r147134 = r147125 * r147133;
        return r147134;
}
↓
double f(double i, double n) {
        double r147135 = n;
        double r147136 = -2.7155205583837284e+82;
        bool r147137 = r147135 <= r147136;
        double r147138 = 100.0;
        double r147139 = i;
        double r147140 = 1.0;
        double r147141 = 0.5;
        double r147142 = 2.0;
        double r147143 = pow(r147139, r147142);
        double r147144 = log(r147140);
        double r147145 = r147144 * r147135;
        double r147146 = fma(r147141, r147143, r147145);
        double r147147 = r147143 * r147144;
        double r147148 = r147141 * r147147;
        double r147149 = r147146 - r147148;
        double r147150 = fma(r147139, r147140, r147149);
        double r147151 = r147150 / r147139;
        double r147152 = r147151 * r147135;
        double r147153 = r147138 * r147152;
        double r147154 = -3.9974032830534776e-251;
        bool r147155 = r147135 <= r147154;
        double r147156 = r147139 / r147135;
        double r147157 = r147140 + r147156;
        double r147158 = pow(r147157, r147135);
        double r147159 = 3.0;
        double r147160 = pow(r147158, r147159);
        double r147161 = pow(r147140, r147159);
        double r147162 = r147160 - r147161;
        double r147163 = r147158 + r147140;
        double r147164 = r147142 * r147135;
        double r147165 = pow(r147157, r147164);
        double r147166 = fma(r147140, r147163, r147165);
        double r147167 = r147166 * r147139;
        double r147168 = r147167 / r147135;
        double r147169 = r147162 / r147168;
        double r147170 = r147138 * r147169;
        double r147171 = 7.185948283350093e-152;
        bool r147172 = r147135 <= r147171;
        double r147173 = 1.0;
        double r147174 = fma(r147144, r147135, r147173);
        double r147175 = fma(r147140, r147139, r147174);
        double r147176 = r147175 - r147140;
        double r147177 = r147176 / r147156;
        double r147178 = r147138 * r147177;
        double r147179 = r147172 ? r147178 : r147153;
        double r147180 = r147155 ? r147170 : r147179;
        double r147181 = r147137 ? r147153 : r147180;
        return r147181;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.4
Target42.3
Herbie22.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -2.7155205583837284e+82 or 7.185948283350093e-152 < n

    1. Initial program 54.6

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

    2. Taylor expanded around 0 40.4

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

    3. Simplified40.4

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

    5. Applied associate-/r/22.3

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

    if -2.7155205583837284e+82 < n < -3.9974032830534776e-251

    1. Initial program 22.0

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

    3. Applied flip3--22.0

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

    4. Applied associate-/l/22.0

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

    5. Simplified22.0

      \[\leadsto 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}}\]

    if -3.9974032830534776e-251 < n < 7.185948283350093e-152

    1. Initial program 33.4

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

    2. Taylor expanded around 0 24.1

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

    3. Simplified24.1

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.71552055838372837 \cdot 10^{82}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -3.9974032830534776 \cdot 10^{-251}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}\\ \mathbf{elif}\;n \le 7.18594828335009282 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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))))