Compound Interest

Average Error: 43.2 → 32.5

Time: 1.1m

Precision: 64

Math

\[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.260760569041402 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}{i} \cdot 100.0\right) \cdot n\\ \mathbf{elif}\;n \le -1.999963205744506:\\ \;\;\;\;100.0 \cdot \frac{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, i \cdot \left(0.5 \cdot i\right) - \left(i \cdot \left(0.5 \cdot i\right)\right) \cdot \log 1.0\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8097259091280393 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)\right)} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)}\right) \cdot 100.0\\ \mathbf{elif}\;n \le 7.680781589380277 \cdot 10^{-180}:\\ \;\;\;\;100.0 \cdot \frac{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, 1\right)\right) - 1.0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100.0 \cdot \frac{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, i \cdot \left(0.5 \cdot i\right) - \left(i \cdot \left(0.5 \cdot i\right)\right) \cdot \log 1.0\right)\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r5401713 = 100.0;
        double r5401714 = 1.0;
        double r5401715 = i;
        double r5401716 = n;
        double r5401717 = r5401715 / r5401716;
        double r5401718 = r5401714 + r5401717;
        double r5401719 = pow(r5401718, r5401716);
        double r5401720 = r5401719 - r5401714;
        double r5401721 = r5401720 / r5401717;
        double r5401722 = r5401713 * r5401721;
        return r5401722;
}

↓

double f(double i, double n) {
        double r5401723 = n;
        double r5401724 = -1.260760569041402e+85;
        bool r5401725 = r5401723 <= r5401724;
        double r5401726 = i;
        double r5401727 = r5401726 / r5401723;
        double r5401728 = 1.0;
        double r5401729 = r5401727 + r5401728;
        double r5401730 = pow(r5401729, r5401723);
        double r5401731 = r5401730 - r5401728;
        double r5401732 = r5401731 / r5401726;
        double r5401733 = 100.0;
        double r5401734 = r5401732 * r5401733;
        double r5401735 = r5401734 * r5401723;
        double r5401736 = -1.999963205744506;
        bool r5401737 = r5401723 <= r5401736;
        double r5401738 = log(r5401728);
        double r5401739 = 0.5;
        double r5401740 = r5401739 * r5401726;
        double r5401741 = r5401726 * r5401740;
        double r5401742 = r5401741 * r5401738;
        double r5401743 = r5401741 - r5401742;
        double r5401744 = fma(r5401726, r5401728, r5401743);
        double r5401745 = fma(r5401738, r5401723, r5401744);
        double r5401746 = r5401745 / r5401727;
        double r5401747 = r5401733 * r5401746;
        double r5401748 = -1.8097259091280393e-258;
        bool r5401749 = r5401723 <= r5401748;
        double r5401750 = exp(r5401731);
        double r5401751 = log(r5401750);
        double r5401752 = cbrt(r5401751);
        double r5401753 = r5401752 * r5401752;
        double r5401754 = cbrt(r5401727);
        double r5401755 = cbrt(r5401754);
        double r5401756 = r5401755 * r5401755;
        double r5401757 = r5401755 * r5401756;
        double r5401758 = r5401754 * r5401757;
        double r5401759 = r5401753 / r5401758;
        double r5401760 = r5401752 / r5401757;
        double r5401761 = r5401759 * r5401760;
        double r5401762 = r5401761 * r5401733;
        double r5401763 = 7.680781589380277e-180;
        bool r5401764 = r5401723 <= r5401763;
        double r5401765 = 1.0;
        double r5401766 = fma(r5401726, r5401728, r5401765);
        double r5401767 = fma(r5401738, r5401723, r5401766);
        double r5401768 = r5401767 - r5401728;
        double r5401769 = r5401768 / r5401727;
        double r5401770 = r5401733 * r5401769;
        double r5401771 = r5401764 ? r5401770 : r5401747;
        double r5401772 = r5401749 ? r5401762 : r5401771;
        double r5401773 = r5401737 ? r5401747 : r5401772;
        double r5401774 = r5401725 ? r5401735 : r5401773;
        return r5401774;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.2
Target	42.9
Herbie	32.5

\[100.0 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1.0 + \frac{i}{n} = 1.0:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1.0 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1.0\right) - 1.0}\\ \end{array}} - 1.0}{\frac{i}{n}}\]

Derivation

1. Split input into 4 regimes

2. if n < -1.260760569041402e+85

   1. Initial program 47.8

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied associate-/r/ 47.4

      \[\leadsto 100.0 \cdot \color{blue}{\left(\frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{i} \cdot n\right)}\]

   4. Applied associate-*r* 47.4

      \[\leadsto \color{blue}{\left(100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{i}\right) \cdot n}\]

   if -1.260760569041402e+85 < n < -1.999963205744506 or 7.680781589380277e-180 < n

   1. Initial program 54.5

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied add-log-exp 54.6

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\frac{i}{n}}\]

   4. Taylor expanded around 0 34.1

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\left(\log 1.0 \cdot n + \left(1.0 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1.0 \cdot {i}^{2}\right)}}{\frac{i}{n}}\]

   5. Simplified 34.1

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, i \cdot \left(i \cdot 0.5\right) - \log 1.0 \cdot \left(i \cdot \left(i \cdot 0.5\right)\right)\right)\right)}}{\frac{i}{n}}\]

   if -1.999963205744506 < n < -1.8097259091280393e-258

   1. Initial program 17.4

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied add-log-exp 17.4

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 17.4

      \[\leadsto 100.0 \cdot \frac{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}{\color{blue}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}}\]

   6. Applied add-cube-cbrt 17.4

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\left(\sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}\]

   7. Applied times-frac 17.4

      \[\leadsto 100.0 \cdot \color{blue}{\left(\frac{\sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}}}\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 17.4

      \[\leadsto 100.0 \cdot \left(\frac{\sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}}}\right)\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 17.4

       \[\leadsto 100.0 \cdot \left(\frac{\sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}\right)}}{\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}}\right)\]

   if -1.8097259091280393e-258 < n < 7.680781589380277e-180

   1. Initial program 30.8

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]

   2. Taylor expanded around 0 20.6

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\left(\log 1.0 \cdot n + \left(1.0 \cdot i + 1\right)\right)} - 1.0}{\frac{i}{n}}\]

   3. Simplified 20.6

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, 1\right)\right)} - 1.0}{\frac{i}{n}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 32.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.260760569041402 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}{i} \cdot 100.0\right) \cdot n\\ \mathbf{elif}\;n \le -1.999963205744506:\\ \;\;\;\;100.0 \cdot \frac{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, i \cdot \left(0.5 \cdot i\right) - \left(i \cdot \left(0.5 \cdot i\right)\right) \cdot \log 1.0\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8097259091280393 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)\right)} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)}\right) \cdot 100.0\\ \mathbf{elif}\;n \le 7.680781589380277 \cdot 10^{-180}:\\ \;\;\;\;100.0 \cdot \frac{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, 1\right)\right) - 1.0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100.0 \cdot \frac{\mathsf{fma}\left(\log 1.0, n, \mathsf{fma}\left(i, 1.0, i \cdot \left(0.5 \cdot i\right) - \left(i \cdot \left(0.5 \cdot i\right)\right) \cdot \log 1.0\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))))