Average Error: 43.4 → 20.5
Time: 25.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(\mathsf{fma}\left(n, \log 1, i \cdot \left(0.5 \cdot i + 1\right)\right) - \left(0.5 \cdot \left(i \cdot i\right)\right) \cdot \log 1\right)\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i \cdot \mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)}{n}}\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\
\;\;\;\;100 \cdot \left(\left(n \cdot \left(\mathsf{fma}\left(n, \log 1, i \cdot \left(0.5 \cdot i + 1\right)\right) - \left(0.5 \cdot \left(i \cdot i\right)\right) \cdot \log 1\right)\right) \cdot \frac{1}{i}\right)\\

\mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\
\;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i \cdot \mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)}{n}}\\

\mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\
\;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r101570 = 100.0;
        double r101571 = 1.0;
        double r101572 = i;
        double r101573 = n;
        double r101574 = r101572 / r101573;
        double r101575 = r101571 + r101574;
        double r101576 = pow(r101575, r101573);
        double r101577 = r101576 - r101571;
        double r101578 = r101577 / r101574;
        double r101579 = r101570 * r101578;
        return r101579;
}
↓
double f(double i, double n) {
        double r101580 = i;
        double r101581 = -0.9893212200252494;
        bool r101582 = r101580 <= r101581;
        double r101583 = 100.0;
        double r101584 = n;
        double r101585 = r101580 / r101584;
        double r101586 = 1.0;
        double r101587 = r101585 + r101586;
        double r101588 = 2.0;
        double r101589 = r101584 * r101588;
        double r101590 = pow(r101587, r101589);
        double r101591 = r101586 * r101586;
        double r101592 = r101590 - r101591;
        double r101593 = pow(r101587, r101584);
        double r101594 = r101593 + r101586;
        double r101595 = r101592 / r101594;
        double r101596 = r101595 / r101585;
        double r101597 = r101583 * r101596;
        double r101598 = 1.5086524363033694e-09;
        bool r101599 = r101580 <= r101598;
        double r101600 = log(r101586);
        double r101601 = 0.5;
        double r101602 = r101601 * r101580;
        double r101603 = r101602 + r101586;
        double r101604 = r101580 * r101603;
        double r101605 = fma(r101584, r101600, r101604);
        double r101606 = r101580 * r101580;
        double r101607 = r101601 * r101606;
        double r101608 = r101607 * r101600;
        double r101609 = r101605 - r101608;
        double r101610 = r101584 * r101609;
        double r101611 = 1.0;
        double r101612 = r101611 / r101580;
        double r101613 = r101610 * r101612;
        double r101614 = r101583 * r101613;
        double r101615 = 8.608246042115979e+235;
        bool r101616 = r101580 <= r101615;
        double r101617 = 3.0;
        double r101618 = pow(r101593, r101617);
        double r101619 = pow(r101586, r101617);
        double r101620 = r101618 - r101619;
        double r101621 = fma(r101586, r101594, r101590);
        double r101622 = r101580 * r101621;
        double r101623 = r101622 / r101584;
        double r101624 = r101620 / r101623;
        double r101625 = r101583 * r101624;
        double r101626 = 1.844738975002478e+296;
        bool r101627 = r101580 <= r101626;
        double r101628 = fma(r101580, r101586, r101611);
        double r101629 = fma(r101584, r101600, r101628);
        double r101630 = r101629 - r101586;
        double r101631 = r101630 / r101585;
        double r101632 = r101631 * r101583;
        double r101633 = r101627 ? r101632 : r101597;
        double r101634 = r101616 ? r101625 : r101633;
        double r101635 = r101599 ? r101614 : r101634;
        double r101636 = r101582 ? r101597 : r101635;
        return r101636;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.4
Target43.3
Herbie20.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -0.9893212200252494 or 1.844738975002478e+296 < i

    1. Initial program 27.9

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

    3. Applied flip--27.9

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

    4. Simplified27.9

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

    5. Simplified27.9

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

    if -0.9893212200252494 < i < 1.5086524363033694e-09

    1. Initial program 51.2

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

    2. Taylor expanded around 0 32.6

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

    3. Simplified32.6

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

    5. Applied div-inv32.7

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

    6. Applied *-un-lft-identity32.7

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

    7. Applied times-frac15.1

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

    8. Simplified15.1

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

    if 1.5086524363033694e-09 < i < 8.608246042115979e+235

    1. Initial program 33.3

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

    3. Applied flip3--33.3

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

    4. Applied associate-/l/33.3

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

    5. Simplified33.3

      \[\leadsto 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\frac{\mathsf{fma}\left(1, 1 + {\left(\frac{i}{n} + 1\right)}^{n}, {\left(\frac{i}{n} + 1\right)}^{\left(2 \cdot n\right)}\right) \cdot i}{n}}}\]

    if 8.608246042115979e+235 < i < 1.844738975002478e+296

    1. Initial program 32.1

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

    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 simplification20.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(\mathsf{fma}\left(n, \log 1, i \cdot \left(0.5 \cdot i + 1\right)\right) - \left(0.5 \cdot \left(i \cdot i\right)\right) \cdot \log 1\right)\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i \cdot \mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)}{n}}\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))))