Average Error: 47.6 → 16.6
Time: 12.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0723466983079063958:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 52622542467923034000:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.0723466983079063958:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 52622542467923034000:\\
\;\;\;\;\left(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)}{i}\right) \cdot n\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\

\end{array}
double f(double i, double n) {
        double r143340 = 100.0;
        double r143341 = 1.0;
        double r143342 = i;
        double r143343 = n;
        double r143344 = r143342 / r143343;
        double r143345 = r143341 + r143344;
        double r143346 = pow(r143345, r143343);
        double r143347 = r143346 - r143341;
        double r143348 = r143347 / r143344;
        double r143349 = r143340 * r143348;
        return r143349;
}
↓
double f(double i, double n) {
        double r143350 = i;
        double r143351 = -0.0723466983079064;
        bool r143352 = r143350 <= r143351;
        double r143353 = 100.0;
        double r143354 = 1.0;
        double r143355 = n;
        double r143356 = r143350 / r143355;
        double r143357 = r143354 + r143356;
        double r143358 = 2.0;
        double r143359 = r143358 * r143355;
        double r143360 = pow(r143357, r143359);
        double r143361 = r143354 * r143354;
        double r143362 = -r143361;
        double r143363 = r143360 + r143362;
        double r143364 = pow(r143357, r143355);
        double r143365 = r143364 + r143354;
        double r143366 = r143363 / r143365;
        double r143367 = r143366 / r143356;
        double r143368 = r143353 * r143367;
        double r143369 = 5.2622542467923034e+19;
        bool r143370 = r143350 <= r143369;
        double r143371 = 0.5;
        double r143372 = pow(r143350, r143358);
        double r143373 = log(r143354);
        double r143374 = r143373 * r143355;
        double r143375 = fma(r143371, r143372, r143374);
        double r143376 = r143372 * r143373;
        double r143377 = r143371 * r143376;
        double r143378 = r143375 - r143377;
        double r143379 = fma(r143350, r143354, r143378);
        double r143380 = r143379 / r143350;
        double r143381 = r143353 * r143380;
        double r143382 = r143381 * r143355;
        double r143383 = r143364 - r143354;
        double r143384 = r143355 / r143350;
        double r143385 = r143383 * r143384;
        double r143386 = r143353 * r143385;
        double r143387 = r143370 ? r143382 : r143386;
        double r143388 = r143352 ? r143368 : r143387;
        return r143388;
}

Error

Bits error versus i

Bits error versus n

Target

Original47.6
Target47.8
Herbie16.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.0723466983079064

    1. Initial program 27.5

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

    3. Applied flip--27.5

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

    4. Simplified27.5

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

    if -0.0723466983079064 < i < 5.2622542467923034e+19

    1. Initial program 58.0

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

    2. Taylor expanded around 0 26.8

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

    3. Simplified26.8

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

    5. Applied associate-/r/10.0

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

    6. Applied associate-*r*10.0

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

    if 5.2622542467923034e+19 < i

    1. Initial program 30.4

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

    3. Applied div-inv30.4

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

    4. Simplified30.4

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0723466983079063958:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 52622542467923034000:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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))))