Average Error: 42.7 → 31.4
Time: 17.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{100 \cdot \frac{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \frac{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\
\;\;\;\;\frac{100 \cdot \frac{\log \left(e^{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\
\;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\
\;\;\;\;100 \cdot \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)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\
\;\;\;\;\frac{\frac{100 \cdot \frac{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 8532543483832934860000:\\
\;\;\;\;100 \cdot \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)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\
\;\;\;\;\frac{\frac{100 \cdot \frac{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\

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

\end{array}
double f(double i, double n) {
        double r228221 = 100.0;
        double r228222 = 1.0;
        double r228223 = i;
        double r228224 = n;
        double r228225 = r228223 / r228224;
        double r228226 = r228222 + r228225;
        double r228227 = pow(r228226, r228224);
        double r228228 = r228227 - r228222;
        double r228229 = r228228 / r228225;
        double r228230 = r228221 * r228229;
        return r228230;
}
↓
double f(double i, double n) {
        double r228231 = i;
        double r228232 = -2.561401665352169e+135;
        bool r228233 = r228231 <= r228232;
        double r228234 = 100.0;
        double r228235 = 1.0;
        double r228236 = -r228235;
        double r228237 = n;
        double r228238 = r228231 / r228237;
        double r228239 = r228235 + r228238;
        double r228240 = 2.0;
        double r228241 = r228240 * r228237;
        double r228242 = pow(r228239, r228241);
        double r228243 = fma(r228236, r228235, r228242);
        double r228244 = exp(r228243);
        double r228245 = log(r228244);
        double r228246 = pow(r228239, r228237);
        double r228247 = r228246 + r228235;
        double r228248 = r228245 / r228247;
        double r228249 = r228234 * r228248;
        double r228250 = r228249 / r228238;
        double r228251 = -1.3992561866449662e-10;
        bool r228252 = r228231 <= r228251;
        double r228253 = pow(r228238, r228237);
        double r228254 = r228253 - r228235;
        double r228255 = r228234 * r228254;
        double r228256 = r228255 / r228238;
        double r228257 = 2.515912909264607e-160;
        bool r228258 = r228231 <= r228257;
        double r228259 = 0.5;
        double r228260 = pow(r228231, r228240);
        double r228261 = log(r228235);
        double r228262 = r228261 * r228237;
        double r228263 = fma(r228259, r228260, r228262);
        double r228264 = r228260 * r228261;
        double r228265 = r228259 * r228264;
        double r228266 = r228263 - r228265;
        double r228267 = fma(r228231, r228235, r228266);
        double r228268 = r228267 / r228238;
        double r228269 = r228234 * r228268;
        double r228270 = 6.02622510223327e-125;
        bool r228271 = r228231 <= r228270;
        double r228272 = r228243 / r228247;
        double r228273 = r228234 * r228272;
        double r228274 = r228273 / r228231;
        double r228275 = 1.0;
        double r228276 = r228275 / r228237;
        double r228277 = r228274 / r228276;
        double r228278 = 8.532543483832935e+21;
        bool r228279 = r228231 <= r228278;
        double r228280 = 7.395559138739582e+219;
        bool r228281 = r228231 <= r228280;
        double r228282 = fma(r228261, r228237, r228275);
        double r228283 = fma(r228235, r228231, r228282);
        double r228284 = r228283 - r228235;
        double r228285 = r228284 / r228238;
        double r228286 = r228234 * r228285;
        double r228287 = r228281 ? r228277 : r228286;
        double r228288 = r228279 ? r228269 : r228287;
        double r228289 = r228271 ? r228277 : r228288;
        double r228290 = r228258 ? r228269 : r228289;
        double r228291 = r228252 ? r228256 : r228290;
        double r228292 = r228233 ? r228250 : r228291;
        return r228292;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.6
Herbie31.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -2.561401665352169e+135

    1. Initial program 15.4

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

    3. Applied associate-*r/15.4

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

    5. Applied flip--15.4

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

    6. Simplified15.4

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

    8. Applied add-log-exp15.4

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

    if -2.561401665352169e+135 < i < -1.3992561866449662e-10

    1. Initial program 41.3

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

    3. Applied associate-*r/41.3

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

    4. Taylor expanded around inf 64.0

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

    5. Simplified27.6

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

    if -1.3992561866449662e-10 < i < 2.515912909264607e-160 or 6.02622510223327e-125 < i < 8.532543483832935e+21

    1. Initial program 49.9

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

    2. Taylor expanded around 0 33.7

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

    3. Simplified33.7

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

    if 2.515912909264607e-160 < i < 6.02622510223327e-125 or 8.532543483832935e+21 < i < 7.395559138739582e+219

    1. Initial program 38.2

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

    3. Applied associate-*r/38.1

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

    5. Applied flip--38.1

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

    6. Simplified38.1

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

    8. Applied div-inv38.1

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

    9. Applied associate-/r*38.0

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

    if 7.395559138739582e+219 < i

    1. Initial program 30.6

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

    2. Taylor expanded around 0 34.7

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

    3. Simplified34.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification31.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.56140166535216897 \cdot 10^{135}:\\ \;\;\;\;\frac{100 \cdot \frac{\log \left(e^{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.3992561866449662 \cdot 10^{-10}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.51591290926460688 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.02622510223326963 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{100 \cdot \frac{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 8532543483832934860000:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.39555913873958208 \cdot 10^{219}:\\ \;\;\;\;\frac{\frac{100 \cdot \frac{\mathsf{fma}\left(-1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))