Average Error: 42.6 → 8.6
Time: 3.0m
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 2.6311580625004606 \cdot 10^{-193}:\\ \;\;\;\;10 \cdot \left(10 \cdot \frac{(e^{n \cdot (e^{\log_* (1 + \log_* (1 + \frac{i}{n}))} - 1)^*} - 1)^*}{\frac{i}{n}}\right)\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 2.1058650875261384 \cdot 10^{-24}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{(e^{i} - 1)^*}{i} \cdot n\right)\\ \end{array}\]
double f(double i, double n) {
        double r33520362 = 100.0;
        double r33520363 = 1.0;
        double r33520364 = i;
        double r33520365 = n;
        double r33520366 = r33520364 / r33520365;
        double r33520367 = r33520363 + r33520366;
        double r33520368 = pow(r33520367, r33520365);
        double r33520369 = r33520368 - r33520363;
        double r33520370 = r33520369 / r33520366;
        double r33520371 = r33520362 * r33520370;
        return r33520371;
}
↓
double f(double i, double n) {
        double r33520372 = i;
        double r33520373 = n;
        double r33520374 = r33520372 / r33520373;
        double r33520375 = 1.0;
        double r33520376 = r33520374 + r33520375;
        double r33520377 = pow(r33520376, r33520373);
        double r33520378 = r33520377 - r33520375;
        double r33520379 = r33520378 / r33520374;
        double r33520380 = 2.6311580625004606e-193;
        bool r33520381 = r33520379 <= r33520380;
        double r33520382 = 10.0;
        double r33520383 = log1p(r33520374);
        double r33520384 = log1p(r33520383);
        double r33520385 = expm1(r33520384);
        double r33520386 = r33520373 * r33520385;
        double r33520387 = expm1(r33520386);
        double r33520388 = r33520387 / r33520374;
        double r33520389 = r33520382 * r33520388;
        double r33520390 = r33520382 * r33520389;
        double r33520391 = 2.1058650875261384e-24;
        bool r33520392 = r33520379 <= r33520391;
        double r33520393 = 100.0;
        double r33520394 = r33520378 / r33520372;
        double r33520395 = r33520393 * r33520394;
        double r33520396 = r33520395 * r33520373;
        double r33520397 = expm1(r33520372);
        double r33520398 = r33520397 / r33520372;
        double r33520399 = r33520398 * r33520373;
        double r33520400 = r33520393 * r33520399;
        double r33520401 = r33520392 ? r33520396 : r33520400;
        double r33520402 = r33520381 ? r33520390 : r33520401;
        return r33520402;
}


100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 2.6311580625004606 \cdot 10^{-193}:\\
\;\;\;\;10 \cdot \left(10 \cdot \frac{(e^{n \cdot (e^{\log_* (1 + \log_* (1 + \frac{i}{n}))} - 1)^*} - 1)^*}{\frac{i}{n}}\right)\\

\mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 2.1058650875261384 \cdot 10^{-24}:\\
\;\;\;\;\left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right) \cdot n\\

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

\end{array}

Error

Bits error versus i

Bits error versus n

Target

Original42.6
Target42.2
Herbie8.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 2.6311580625004606e-193

    1. Initial program 39.6

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

    3. Applied pow-to-exp40.0

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

    4. Applied expm1-def32.0

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

    5. Simplified7.3

      \[\leadsto 100 \cdot \frac{(e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1)^*}{\frac{i}{n}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt7.3

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

    8. Applied associate-*l*7.4

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

    10. Applied expm1-log1p-u7.5

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

    if 2.6311580625004606e-193 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 2.1058650875261384e-24

    1. Initial program 1.9

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

    3. Applied associate-/r/1.9

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

    4. Applied associate-*r*1.9

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

    if 2.1058650875261384e-24 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))

    1. Initial program 59.1

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

    3. Applied pow-to-exp60.5

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

    4. Applied expm1-def60.5

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

    5. Simplified60.5

      \[\leadsto 100 \cdot \frac{(e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1)^*}{\frac{i}{n}}\]

    6. Taylor expanded around inf 61.5

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

    7. Simplified13.5

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 2.6311580625004606 \cdot 10^{-193}:\\ \;\;\;\;10 \cdot \left(10 \cdot \frac{(e^{n \cdot (e^{\log_* (1 + \log_* (1 + \frac{i}{n}))} - 1)^*} - 1)^*}{\frac{i}{n}}\right)\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 2.1058650875261384 \cdot 10^{-24}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{(e^{i} - 1)^*}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(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))))