Average Error: 42.1 → 17.9
Time: 35.4s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -310.361827324758:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 10.814933530045678:\\ \;\;\;\;\left(\left(100 + i \cdot 50\right) + i \cdot \left(i \cdot \frac{50}{3}\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right) \cdot n\right) \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -310.361827324758:\\
\;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 10.814933530045678:\\
\;\;\;\;\left(\left(100 + i \cdot 50\right) + i \cdot \left(i \cdot \frac{50}{3}\right)\right) \cdot n\\

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

\end{array}
double f(double i, double n) {
        double r6229380 = 100.0;
        double r6229381 = 1.0;
        double r6229382 = i;
        double r6229383 = n;
        double r6229384 = r6229382 / r6229383;
        double r6229385 = r6229381 + r6229384;
        double r6229386 = pow(r6229385, r6229383);
        double r6229387 = r6229386 - r6229381;
        double r6229388 = r6229387 / r6229384;
        double r6229389 = r6229380 * r6229388;
        return r6229389;
}
↓
double f(double i, double n) {
        double r6229390 = i;
        double r6229391 = -310.361827324758;
        bool r6229392 = r6229390 <= r6229391;
        double r6229393 = 100.0;
        double r6229394 = n;
        double r6229395 = r6229390 / r6229394;
        double r6229396 = 1.0;
        double r6229397 = r6229395 + r6229396;
        double r6229398 = pow(r6229397, r6229394);
        double r6229399 = r6229398 / r6229395;
        double r6229400 = /* ERROR: no posit support in C */;
        double r6229401 = /* ERROR: no posit support in C */;
        double r6229402 = r6229396 / r6229395;
        double r6229403 = r6229401 - r6229402;
        double r6229404 = r6229393 * r6229403;
        double r6229405 = 10.814933530045678;
        bool r6229406 = r6229390 <= r6229405;
        double r6229407 = 50.0;
        double r6229408 = r6229390 * r6229407;
        double r6229409 = r6229393 + r6229408;
        double r6229410 = 16.666666666666668;
        double r6229411 = r6229390 * r6229410;
        double r6229412 = r6229390 * r6229411;
        double r6229413 = r6229409 + r6229412;
        double r6229414 = r6229413 * r6229394;
        double r6229415 = r6229398 / r6229390;
        double r6229416 = r6229396 / r6229390;
        double r6229417 = r6229415 - r6229416;
        double r6229418 = r6229417 * r6229394;
        double r6229419 = r6229418 * r6229393;
        double r6229420 = r6229406 ? r6229414 : r6229419;
        double r6229421 = r6229392 ? r6229404 : r6229420;
        return r6229421;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.1
Target42.2
Herbie17.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -310.361827324758

    1. Initial program 27.2

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

    3. Applied div-sub27.2

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

    5. Applied insert-posit1613.0

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

    if -310.361827324758 < i < 10.814933530045678

    1. Initial program 50.0

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

    3. Applied div-sub50.0

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

    4. Taylor expanded around 0 17.1

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right) + \left(n + \frac{1}{2} \cdot \left(i \cdot n\right)\right)\right)}\]

    5. Simplified17.1

      \[\leadsto 100 \cdot \color{blue}{\left(\left(n + \frac{1}{6} \cdot \left(n \cdot \left(i \cdot i\right)\right)\right) + i \cdot \left(n \cdot \frac{1}{2}\right)\right)}\]

    6. Taylor expanded around -inf 17.1

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right) + \left(n + \frac{1}{2} \cdot \left(i \cdot n\right)\right)\right)}\]

    7. Simplified17.1

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(n \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right)}\]

    8. Taylor expanded around -inf 17.1

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

    9. Simplified17.1

      \[\leadsto \color{blue}{n \cdot \left(\left(i \cdot \frac{50}{3}\right) \cdot i + \left(100 + i \cdot 50\right)\right)}\]

    if 10.814933530045678 < i

    1. Initial program 30.5

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

    3. Applied div-sub30.5

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

    5. Applied associate-/r/34.0

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

    6. Applied associate-/r/30.5

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

    7. Applied distribute-rgt-out--30.5

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -310.361827324758:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 10.814933530045678:\\ \;\;\;\;\left(\left(100 + i \cdot 50\right) + i \cdot \left(i \cdot \frac{50}{3}\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right) \cdot n\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (i n)
  :name "Compound Interest"

  :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))))