Average Error: 47.6 → 16.6
Time: 12.6s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0723466983079063958:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 52622542467923034000:\\ \;\;\;\;100 \cdot \left(\left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \end{array}\]
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:\\
\;\;\;\;100 \cdot \left(\left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right) \cdot n\right)\\

\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 r152343 = 100.0;
        double r152344 = 1.0;
        double r152345 = i;
        double r152346 = n;
        double r152347 = r152345 / r152346;
        double r152348 = r152344 + r152347;
        double r152349 = pow(r152348, r152346);
        double r152350 = r152349 - r152344;
        double r152351 = r152350 / r152347;
        double r152352 = r152343 * r152351;
        return r152352;
}
↓
double f(double i, double n) {
        double r152353 = i;
        double r152354 = -0.0723466983079064;
        bool r152355 = r152353 <= r152354;
        double r152356 = 100.0;
        double r152357 = 1.0;
        double r152358 = n;
        double r152359 = r152353 / r152358;
        double r152360 = r152357 + r152359;
        double r152361 = 2.0;
        double r152362 = r152361 * r152358;
        double r152363 = pow(r152360, r152362);
        double r152364 = r152357 * r152357;
        double r152365 = -r152364;
        double r152366 = r152363 + r152365;
        double r152367 = pow(r152360, r152358);
        double r152368 = r152367 + r152357;
        double r152369 = r152366 / r152368;
        double r152370 = r152369 / r152359;
        double r152371 = r152356 * r152370;
        double r152372 = 5.2622542467923034e+19;
        bool r152373 = r152353 <= r152372;
        double r152374 = 0.5;
        double r152375 = r152374 * r152353;
        double r152376 = log(r152357);
        double r152377 = r152376 * r152358;
        double r152378 = r152377 / r152353;
        double r152379 = r152378 + r152357;
        double r152380 = r152375 + r152379;
        double r152381 = r152353 * r152376;
        double r152382 = r152374 * r152381;
        double r152383 = r152380 - r152382;
        double r152384 = r152383 * r152358;
        double r152385 = r152356 * r152384;
        double r152386 = r152367 - r152357;
        double r152387 = r152358 / r152353;
        double r152388 = r152386 * r152387;
        double r152389 = r152356 * r152388;
        double r152390 = r152373 ? r152385 : r152389;
        double r152391 = r152355 ? r152371 : r152390;
        return r152391;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

    4. Applied associate-/r/10.0

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

    6. Applied add-log-exp10.3

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

    7. Taylor expanded around 0 10.0

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

    8. Simplified10.0

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

    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

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

Reproduce

herbie shell --seed 2020064 
(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))))