Average Error: 43.1 → 23.6
Time: 19.4s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)}{i}\\ \mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -1.0912719332224593 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.25395219693272548 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\
\;\;\;\;\left(100 \cdot \frac{\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}\right) \cdot n\\

\mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\
\;\;\;\;\frac{100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)}{i}\\

\mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\
\;\;\;\;\left(100 \cdot \frac{\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}\right) \cdot n\\

\mathbf{elif}\;n \le -1.0912719332224593 \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:\\
\;\;\;\;\left(100 \cdot \frac{\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}\right) \cdot 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 6.25395219693272548 \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}:\\
\;\;\;\;\left(100 \cdot \frac{\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}\right) \cdot n\\

\end{array}
double f(double i, double n) {
        double r171170 = 100.0;
        double r171171 = 1.0;
        double r171172 = i;
        double r171173 = n;
        double r171174 = r171172 / r171173;
        double r171175 = r171171 + r171174;
        double r171176 = pow(r171175, r171173);
        double r171177 = r171176 - r171171;
        double r171178 = r171177 / r171174;
        double r171179 = r171170 * r171178;
        return r171179;
}
↓
double f(double i, double n) {
        double r171180 = n;
        double r171181 = -1.0029723224091834e+174;
        bool r171182 = r171180 <= r171181;
        double r171183 = 100.0;
        double r171184 = 1.0;
        double r171185 = i;
        double r171186 = 0.5;
        double r171187 = r171185 * r171185;
        double r171188 = log(r171184);
        double r171189 = r171188 * r171180;
        double r171190 = fma(r171186, r171187, r171189);
        double r171191 = fma(r171184, r171185, r171190);
        double r171192 = 2.0;
        double r171193 = pow(r171185, r171192);
        double r171194 = r171193 * r171188;
        double r171195 = r171186 * r171194;
        double r171196 = r171191 - r171195;
        double r171197 = r171196 / r171185;
        double r171198 = r171183 * r171197;
        double r171199 = r171198 * r171180;
        double r171200 = -4.0369911640829986e+85;
        bool r171201 = r171180 <= r171200;
        double r171202 = r171185 / r171180;
        double r171203 = r171184 + r171202;
        double r171204 = pow(r171203, r171180);
        double r171205 = r171204 - r171184;
        double r171206 = r171205 * r171180;
        double r171207 = r171183 * r171206;
        double r171208 = r171207 / r171185;
        double r171209 = -3.317476088456595e+63;
        bool r171210 = r171180 <= r171209;
        double r171211 = -1.0912719332224593e+26;
        bool r171212 = r171180 <= r171211;
        double r171213 = r171183 * r171205;
        double r171214 = r171213 / r171202;
        double r171215 = -934893.9033682138;
        bool r171216 = r171180 <= r171215;
        double r171217 = 1.4472641457797862e-291;
        bool r171218 = r171180 <= r171217;
        double r171219 = r171204 / r171202;
        double r171220 = r171184 / r171202;
        double r171221 = r171219 - r171220;
        double r171222 = r171183 * r171221;
        double r171223 = 6.2539521969327255e-155;
        bool r171224 = r171180 <= r171223;
        double r171225 = 1.0;
        double r171226 = fma(r171188, r171180, r171225);
        double r171227 = fma(r171184, r171185, r171226);
        double r171228 = r171227 - r171184;
        double r171229 = r171228 / r171202;
        double r171230 = r171183 * r171229;
        double r171231 = r171224 ? r171230 : r171199;
        double r171232 = r171218 ? r171222 : r171231;
        double r171233 = r171216 ? r171199 : r171232;
        double r171234 = r171212 ? r171214 : r171233;
        double r171235 = r171210 ? r171199 : r171234;
        double r171236 = r171201 ? r171208 : r171235;
        double r171237 = r171182 ? r171199 : r171236;
        return r171237;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.1
Target43.0
Herbie23.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if n < -1.0029723224091834e+174 or -4.0369911640829986e+85 < n < -3.317476088456595e+63 or -1.0912719332224593e+26 < n < -934893.9033682138 or 6.2539521969327255e-155 < n

    1. Initial program 56.2

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

    3. Applied associate-/r/55.9

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

    4. Taylor expanded around 0 21.3

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

    5. Simplified21.3

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

    7. Applied associate-*r*21.3

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

    if -1.0029723224091834e+174 < n < -4.0369911640829986e+85

    1. Initial program 40.6

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

    3. Applied associate-/r/40.3

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

    5. Applied associate-*l/40.2

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

    6. Applied associate-*r/40.2

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

    if -3.317476088456595e+63 < n < -1.0912719332224593e+26

    1. Initial program 30.6

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

    3. Applied associate-*r/30.5

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

    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 < 6.2539521969327255e-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 5 regimes into one program.

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\ \;\;\;\;\frac{100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)}{i}\\ \mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -1.0912719332224593 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 1.4472641457797862 \cdot 10^{-291}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.25395219693272548 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]

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))))