Average Error: 42.9 → 22.2
Time: 35.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -7.524185979796087207979591237813621340536 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.943238893158424573925913136918097734451:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 1.540498293983815894197419121524134450797 \cdot 10^{208}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -7.524185979796087207979591237813621340536 \cdot 10^{-8}:\\
\;\;\;\;100 \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 1.943238893158424573925913136918097734451:\\
\;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot n\right)\\

\mathbf{elif}\;i \le 1.540498293983815894197419121524134450797 \cdot 10^{208}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

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

\end{array}
double f(double i, double n) {
        double r5648535 = 100.0;
        double r5648536 = 1.0;
        double r5648537 = i;
        double r5648538 = n;
        double r5648539 = r5648537 / r5648538;
        double r5648540 = r5648536 + r5648539;
        double r5648541 = pow(r5648540, r5648538);
        double r5648542 = r5648541 - r5648536;
        double r5648543 = r5648542 / r5648539;
        double r5648544 = r5648535 * r5648543;
        return r5648544;
}
↓
double f(double i, double n) {
        double r5648545 = i;
        double r5648546 = -7.524185979796087e-08;
        bool r5648547 = r5648545 <= r5648546;
        double r5648548 = 100.0;
        double r5648549 = n;
        double r5648550 = r5648545 / r5648549;
        double r5648551 = 1.0;
        double r5648552 = r5648550 + r5648551;
        double r5648553 = pow(r5648552, r5648549);
        double r5648554 = r5648553 * r5648553;
        double r5648555 = r5648554 * r5648553;
        double r5648556 = r5648551 * r5648551;
        double r5648557 = r5648556 * r5648551;
        double r5648558 = r5648555 - r5648557;
        double r5648559 = r5648553 + r5648551;
        double r5648560 = fma(r5648551, r5648559, r5648554);
        double r5648561 = r5648558 / r5648560;
        double r5648562 = r5648561 / r5648550;
        double r5648563 = r5648548 * r5648562;
        double r5648564 = 1.9432388931584246;
        bool r5648565 = r5648545 <= r5648564;
        double r5648566 = log(r5648551);
        double r5648567 = r5648545 * r5648545;
        double r5648568 = 0.5;
        double r5648569 = r5648545 * r5648551;
        double r5648570 = fma(r5648567, r5648568, r5648569);
        double r5648571 = fma(r5648549, r5648566, r5648570);
        double r5648572 = r5648568 * r5648566;
        double r5648573 = r5648572 * r5648567;
        double r5648574 = r5648571 - r5648573;
        double r5648575 = r5648574 / r5648545;
        double r5648576 = r5648575 * r5648549;
        double r5648577 = r5648548 * r5648576;
        double r5648578 = 1.540498293983816e+208;
        bool r5648579 = r5648545 <= r5648578;
        double r5648580 = r5648553 / r5648550;
        double r5648581 = r5648551 / r5648550;
        double r5648582 = r5648580 - r5648581;
        double r5648583 = r5648548 * r5648582;
        double r5648584 = 1.0;
        double r5648585 = fma(r5648545, r5648551, r5648584);
        double r5648586 = fma(r5648549, r5648566, r5648585);
        double r5648587 = r5648586 - r5648551;
        double r5648588 = r5648587 / r5648550;
        double r5648589 = r5648588 * r5648548;
        double r5648590 = r5648579 ? r5648583 : r5648589;
        double r5648591 = r5648565 ? r5648577 : r5648590;
        double r5648592 = r5648547 ? r5648563 : r5648591;
        return r5648592;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.9
Target42.1
Herbie22.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -7.524185979796087e-08

    1. Initial program 30.2

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

    3. Applied flip3--30.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 \cdot 1 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]

    4. Simplified30.2

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

    5. Simplified30.2

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

    if -7.524185979796087e-08 < i < 1.9432388931584246

    1. Initial program 50.4

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

    2. Taylor expanded around 0 34.1

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

    3. Simplified34.1

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

    5. Applied associate-/r/17.0

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

    if 1.9432388931584246 < i < 1.540498293983816e+208

    1. Initial program 30.7

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

    3. Applied div-sub30.7

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

    if 1.540498293983816e+208 < i

    1. Initial program 31.8

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

    2. Taylor expanded around 0 34.0

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

    3. Simplified34.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -7.524185979796087207979591237813621340536 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \left(1 \cdot 1\right) \cdot 1}{\mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.943238893158424573925913136918097734451:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 1.540498293983815894197419121524134450797 \cdot 10^{208}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))