Average Error: 42.7 → 13.3
Time: 26.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -5.737712731949101 \cdot 10^{-05}:\\ \;\;\;\;\frac{1}{\frac{i}{n \cdot \mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}}\\ \mathbf{elif}\;i \le 0.0004866774544189746:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{i}{n}, \frac{\frac{1}{200}}{n}, \mathsf{fma}\left(\frac{i}{n}, \frac{-1}{200}, \frac{\frac{1}{100}}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -5.737712731949101 \cdot 10^{-05}:\\
\;\;\;\;\frac{1}{\frac{i}{n \cdot \mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}}\\

\mathbf{elif}\;i \le 0.0004866774544189746:\\
\;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{i}{n}, \frac{\frac{1}{200}}{n}, \mathsf{fma}\left(\frac{i}{n}, \frac{-1}{200}, \frac{\frac{1}{100}}{n}\right)\right)}\\

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

\end{array}
double f(double i, double n) {
        double r5183450 = 100.0;
        double r5183451 = 1.0;
        double r5183452 = i;
        double r5183453 = n;
        double r5183454 = r5183452 / r5183453;
        double r5183455 = r5183451 + r5183454;
        double r5183456 = pow(r5183455, r5183453);
        double r5183457 = r5183456 - r5183451;
        double r5183458 = r5183457 / r5183454;
        double r5183459 = r5183450 * r5183458;
        return r5183459;
}
↓
double f(double i, double n) {
        double r5183460 = i;
        double r5183461 = -5.737712731949101e-05;
        bool r5183462 = r5183460 <= r5183461;
        double r5183463 = 1.0;
        double r5183464 = n;
        double r5183465 = 100.0;
        double r5183466 = r5183460 / r5183464;
        double r5183467 = log1p(r5183466);
        double r5183468 = r5183467 * r5183464;
        double r5183469 = exp(r5183468);
        double r5183470 = -100.0;
        double r5183471 = fma(r5183465, r5183469, r5183470);
        double r5183472 = r5183464 * r5183471;
        double r5183473 = r5183460 / r5183472;
        double r5183474 = r5183463 / r5183473;
        double r5183475 = 0.0004866774544189746;
        bool r5183476 = r5183460 <= r5183475;
        double r5183477 = 0.005;
        double r5183478 = r5183477 / r5183464;
        double r5183479 = -0.005;
        double r5183480 = 0.01;
        double r5183481 = r5183480 / r5183464;
        double r5183482 = fma(r5183466, r5183479, r5183481);
        double r5183483 = fma(r5183466, r5183478, r5183482);
        double r5183484 = r5183463 / r5183483;
        double r5183485 = r5183463 + r5183466;
        double r5183486 = pow(r5183485, r5183464);
        double r5183487 = r5183465 * r5183486;
        double r5183488 = r5183487 + r5183470;
        double r5183489 = r5183488 / r5183466;
        double r5183490 = r5183476 ? r5183484 : r5183489;
        double r5183491 = r5183462 ? r5183474 : r5183490;
        return r5183491;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.2
Herbie13.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -5.737712731949101e-05

    1. Initial program 29.1

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

    2. Simplified29.1

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

    4. Applied add-exp-log29.1

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

    5. Applied pow-exp29.1

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

    6. Simplified5.7

      \[\leadsto \frac{\mathsf{fma}\left(100, e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}, -100\right)}{\frac{i}{n}}\]
    7. Using strategy rm

    8. Applied clear-num5.8

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}}}\]
    9. Using strategy rm

    10. Applied associate-/l/6.2

      \[\leadsto \frac{1}{\color{blue}{\frac{i}{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right) \cdot n}}}\]

    if -5.737712731949101e-05 < i < 0.0004866774544189746

    1. Initial program 50.1

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

    2. Simplified50.1

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

    4. Applied add-exp-log50.1

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

    5. Applied pow-exp50.1

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

    6. Simplified49.3

      \[\leadsto \frac{\mathsf{fma}\left(100, e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}, -100\right)}{\frac{i}{n}}\]
    7. Using strategy rm

    8. Applied clear-num49.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}}}\]

    9. Taylor expanded around 0 13.3

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{200} \cdot \frac{i}{{n}^{2}} + \frac{1}{100} \cdot \frac{1}{n}\right) - \frac{1}{200} \cdot \frac{i}{n}}}\]

    10. Simplified13.2

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{i}{n}, \frac{-1}{200}, \mathsf{fma}\left(\frac{i}{n \cdot n}, \frac{1}{200}, \frac{\frac{1}{100}}{n}\right)\right)}}\]
    11. Using strategy rm

    12. Applied clear-num13.2

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{i}{n}, \frac{-1}{200}, \mathsf{fma}\left(\frac{i}{n \cdot n}, \frac{1}{200}, \frac{\frac{1}{100}}{n}\right)\right)}{1}}}\]

    13. Simplified11.9

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{i}{n}, \frac{\frac{1}{200}}{n}, \mathsf{fma}\left(\frac{i}{n}, \frac{-1}{200}, \frac{\frac{1}{100}}{n}\right)\right)}}\]

    if 0.0004866774544189746 < i

    1. Initial program 31.4

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

    2. Simplified31.4

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

    4. Applied fma-udef31.4

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -5.737712731949101 \cdot 10^{-05}:\\ \;\;\;\;\frac{1}{\frac{i}{n \cdot \mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}}\\ \mathbf{elif}\;i \le 0.0004866774544189746:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{i}{n}, \frac{\frac{1}{200}}{n}, \mathsf{fma}\left(\frac{i}{n}, \frac{-1}{200}, \frac{\frac{1}{100}}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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))))