Average Error: 43.1 → 21.0
Time: 21.4s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.346652527723265269088415152509696781635:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 5.321174585513155719949917669658167093164 \cdot 10^{-285}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.089742171439268069923790151154466595501 \cdot 10^{-181}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.346652527723265269088415152509696781635:\\
\;\;\;\;\left(100 \cdot \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}\right) \cdot n\\

\mathbf{elif}\;n \le 5.321174585513155719949917669658167093164 \cdot 10^{-285}:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le 3.089742171439268069923790151154466595501 \cdot 10^{-181}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;\left(100 \cdot \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}\right) \cdot n\\

\end{array}
double f(double i, double n) {
        double r139483 = 100.0;
        double r139484 = 1.0;
        double r139485 = i;
        double r139486 = n;
        double r139487 = r139485 / r139486;
        double r139488 = r139484 + r139487;
        double r139489 = pow(r139488, r139486);
        double r139490 = r139489 - r139484;
        double r139491 = r139490 / r139487;
        double r139492 = r139483 * r139491;
        return r139492;
}
↓
double f(double i, double n) {
        double r139493 = n;
        double r139494 = -2.3466525277232653;
        bool r139495 = r139493 <= r139494;
        double r139496 = 100.0;
        double r139497 = 1.0;
        double r139498 = i;
        double r139499 = r139497 * r139498;
        double r139500 = 0.5;
        double r139501 = 2.0;
        double r139502 = pow(r139498, r139501);
        double r139503 = r139500 * r139502;
        double r139504 = log(r139497);
        double r139505 = r139504 * r139493;
        double r139506 = r139503 + r139505;
        double r139507 = r139499 + r139506;
        double r139508 = r139502 * r139504;
        double r139509 = r139500 * r139508;
        double r139510 = r139507 - r139509;
        double r139511 = r139510 / r139498;
        double r139512 = r139496 * r139511;
        double r139513 = r139512 * r139493;
        double r139514 = 5.321174585513156e-285;
        bool r139515 = r139493 <= r139514;
        double r139516 = r139498 / r139493;
        double r139517 = r139497 + r139516;
        double r139518 = pow(r139517, r139493);
        double r139519 = r139518 - r139497;
        double r139520 = r139496 * r139519;
        double r139521 = r139520 / r139516;
        double r139522 = 3.089742171439268e-181;
        bool r139523 = r139493 <= r139522;
        double r139524 = 1.0;
        double r139525 = r139505 + r139524;
        double r139526 = r139499 + r139525;
        double r139527 = r139526 - r139497;
        double r139528 = r139527 / r139516;
        double r139529 = r139496 * r139528;
        double r139530 = r139523 ? r139529 : r139513;
        double r139531 = r139515 ? r139521 : r139530;
        double r139532 = r139495 ? r139513 : r139531;
        return r139532;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.1
Target43.3
Herbie21.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -2.3466525277232653 or 3.089742171439268e-181 < n

    1. Initial program 52.9

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

    2. Taylor expanded around 0 39.0

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

    4. Applied associate-/r/22.3

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

    5. Applied associate-*r*22.3

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

    if -2.3466525277232653 < n < 5.321174585513156e-285

    1. Initial program 15.2

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

    3. Applied associate-*r/15.2

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

    if 5.321174585513156e-285 < n < 3.089742171439268e-181

    1. Initial program 41.9

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

    2. Taylor expanded around 0 27.6

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

  4. Final simplification21.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.346652527723265269088415152509696781635:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 5.321174585513155719949917669658167093164 \cdot 10^{-285}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.089742171439268069923790151154466595501 \cdot 10^{-181}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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))))