Compound Interest

Average Error: 43.2 → 21.6

Time: 30.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -587178759827394.125:\\ \;\;\;\;\frac{\mathsf{fma}\left(100, {\left(\frac{1}{n} \cdot i\right)}^{n}, -100\right)}{i} \cdot n\\ \mathbf{elif}\;i \le 1.839542617760591374384239285215744794771 \cdot 10^{-53}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \log \left(e^{\left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)}\right)\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 3.511545104334265421804014757584958652582 \cdot 10^{-14}:\\ \;\;\;\;\left(\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n} \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\right) \cdot \log \left(e^{\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}}\right)\\ \mathbf{elif}\;i \le 0.01027192758964808570742999194180811173283:\\ \;\;\;\;\left(\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n} \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\right) \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\\ \mathbf{elif}\;i \le 7.312595874998127847724356328504784603445 \cdot 10^{224} \lor \neg \left(i \le 1.674538242776564142877968073717764571719 \cdot 10^{263}\right):\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r118692 = 100.0;
        double r118693 = 1.0;
        double r118694 = i;
        double r118695 = n;
        double r118696 = r118694 / r118695;
        double r118697 = r118693 + r118696;
        double r118698 = pow(r118697, r118695);
        double r118699 = r118698 - r118693;
        double r118700 = r118699 / r118696;
        double r118701 = r118692 * r118700;
        return r118701;
}

↓

double f(double i, double n) {
        double r118702 = i;
        double r118703 = -587178759827394.1;
        bool r118704 = r118702 <= r118703;
        double r118705 = 100.0;
        double r118706 = 1.0;
        double r118707 = n;
        double r118708 = r118706 / r118707;
        double r118709 = r118708 * r118702;
        double r118710 = pow(r118709, r118707);
        double r118711 = -r118705;
        double r118712 = fma(r118705, r118710, r118711);
        double r118713 = r118712 / r118702;
        double r118714 = r118713 * r118707;
        double r118715 = 1.8395426177605914e-53;
        bool r118716 = r118702 <= r118715;
        double r118717 = 1.0;
        double r118718 = log(r118717);
        double r118719 = r118702 * r118702;
        double r118720 = 0.5;
        double r118721 = r118720 * r118718;
        double r118722 = r118720 - r118721;
        double r118723 = r118719 * r118722;
        double r118724 = exp(r118723);
        double r118725 = log(r118724);
        double r118726 = fma(r118707, r118718, r118725);
        double r118727 = fma(r118717, r118702, r118726);
        double r118728 = r118727 / r118702;
        double r118729 = r118705 * r118728;
        double r118730 = r118729 * r118707;
        double r118731 = 3.5115451043342654e-14;
        bool r118732 = r118702 <= r118731;
        double r118733 = fma(r118707, r118718, r118723);
        double r118734 = fma(r118717, r118702, r118733);
        double r118735 = r118734 / r118702;
        double r118736 = r118705 * r118735;
        double r118737 = r118736 * r118707;
        double r118738 = cbrt(r118737);
        double r118739 = r118738 * r118738;
        double r118740 = exp(r118738);
        double r118741 = log(r118740);
        double r118742 = r118739 * r118741;
        double r118743 = 0.010271927589648086;
        bool r118744 = r118702 <= r118743;
        double r118745 = r118739 * r118738;
        double r118746 = 7.312595874998128e+224;
        bool r118747 = r118702 <= r118746;
        double r118748 = 1.6745382427765641e+263;
        bool r118749 = r118702 <= r118748;
        double r118750 = !r118749;
        bool r118751 = r118747 || r118750;
        double r118752 = r118702 / r118707;
        double r118753 = r118717 + r118752;
        double r118754 = pow(r118753, r118707);
        double r118755 = 3.0;
        double r118756 = pow(r118754, r118755);
        double r118757 = pow(r118717, r118755);
        double r118758 = r118756 - r118757;
        double r118759 = r118754 + r118717;
        double r118760 = 2.0;
        double r118761 = r118760 * r118707;
        double r118762 = pow(r118753, r118761);
        double r118763 = fma(r118717, r118759, r118762);
        double r118764 = r118758 / r118763;
        double r118765 = r118764 / r118702;
        double r118766 = r118705 * r118765;
        double r118767 = r118766 * r118707;
        double r118768 = fma(r118718, r118707, r118706);
        double r118769 = fma(r118717, r118702, r118768);
        double r118770 = r118769 - r118717;
        double r118771 = r118770 / r118752;
        double r118772 = r118705 * r118771;
        double r118773 = r118751 ? r118767 : r118772;
        double r118774 = r118744 ? r118745 : r118773;
        double r118775 = r118732 ? r118742 : r118774;
        double r118776 = r118716 ? r118730 : r118775;
        double r118777 = r118704 ? r118714 : r118776;
        return r118777;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.2
Target	42.5
Herbie	21.6

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

Derivation

1. Split input into 6 regimes

2. if i < -587178759827394.1

   1. Initial program 27.7

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

   3. Applied associate-/r/ 28.3

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

   4. Applied associate-*r* 28.3

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

   5. Taylor expanded around inf 64.0

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

   6. Simplified 19.1

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

   if -587178759827394.1 < i < 1.8395426177605914e-53

   1. Initial program 50.4

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

   3. Applied associate-/r/ 50.1

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

   4. Applied associate-*r* 50.1

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

   5. Taylor expanded around 0 17.7

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

   6. Simplified 17.7

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

   8. Applied add-log-exp 17.8

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

   if 1.8395426177605914e-53 < i < 3.5115451043342654e-14

   1. Initial program 53.8

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

   3. Applied associate-/r/ 53.7

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

   4. Applied associate-*r* 53.7

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

   5. Taylor expanded around 0 17.8

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

   6. Simplified 17.8

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

   8. Applied add-cube-cbrt 18.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n} \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\right) \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}}\]
   9. Using strategy rm

   10. Applied add-log-exp 52.2

       \[\leadsto \left(\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n} \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\right) \cdot \color{blue}{\log \left(e^{\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}}\right)}\]

   if 3.5115451043342654e-14 < i < 0.010271927589648086

   1. Initial program 51.1

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

   3. Applied associate-/r/ 51.1

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

   4. Applied associate-*r* 51.1

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

   5. Taylor expanded around 0 21.3

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

   6. Simplified 21.3

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

   8. Applied add-cube-cbrt 21.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n} \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\right) \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}}\]

   if 0.010271927589648086 < i < 7.312595874998128e+224 or 1.6745382427765641e+263 < i

   1. Initial program 32.9

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

   3. Applied associate-/r/ 32.9

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

   4. Applied associate-*r* 32.9

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

   6. Applied flip3-- 33.0

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

   7. Simplified 33.0

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

   if 7.312595874998128e+224 < i < 1.6745382427765641e+263

   1. Initial program 33.4

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

   2. Taylor expanded around 0 33.7

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

   3. Simplified 33.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 21.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -587178759827394.125:\\ \;\;\;\;\frac{\mathsf{fma}\left(100, {\left(\frac{1}{n} \cdot i\right)}^{n}, -100\right)}{i} \cdot n\\ \mathbf{elif}\;i \le 1.839542617760591374384239285215744794771 \cdot 10^{-53}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \log \left(e^{\left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)}\right)\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;i \le 3.511545104334265421804014757584958652582 \cdot 10^{-14}:\\ \;\;\;\;\left(\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n} \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\right) \cdot \log \left(e^{\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}}\right)\\ \mathbf{elif}\;i \le 0.01027192758964808570742999194180811173283:\\ \;\;\;\;\left(\sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n} \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\right) \cdot \sqrt[3]{\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n}\\ \mathbf{elif}\;i \le 7.312595874998127847724356328504784603445 \cdot 10^{224} \lor \neg \left(i \le 1.674538242776564142877968073717764571719 \cdot 10^{263}\right):\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))