Compound Interest

Average Error: 43.0 → 23.5

Time: 27.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -2.260047491173430560879176932933542305034 \cdot 10^{95}:\\ \;\;\;\;100 \cdot \frac{\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;n \le -3.040807945417772148751844957499095808404 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.48872624183818144300366844690655908205 \cdot 10^{-106}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(100 \cdot \left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right)\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{\sqrt[3]{n}}}\\ \end{array}\]

C

double f(double i, double n) {
        double r103693 = 100.0;
        double r103694 = 1.0;
        double r103695 = i;
        double r103696 = n;
        double r103697 = r103695 / r103696;
        double r103698 = r103694 + r103697;
        double r103699 = pow(r103698, r103696);
        double r103700 = r103699 - r103694;
        double r103701 = r103700 / r103697;
        double r103702 = r103693 * r103701;
        return r103702;
}

↓

double f(double i, double n) {
        double r103703 = n;
        double r103704 = -2.2600474911734306e+95;
        bool r103705 = r103703 <= r103704;
        double r103706 = 100.0;
        double r103707 = 1.0;
        double r103708 = i;
        double r103709 = 0.5;
        double r103710 = 2.0;
        double r103711 = pow(r103708, r103710);
        double r103712 = log(r103707);
        double r103713 = r103712 * r103703;
        double r103714 = fma(r103709, r103711, r103713);
        double r103715 = fma(r103707, r103708, r103714);
        double r103716 = r103711 * r103712;
        double r103717 = r103709 * r103716;
        double r103718 = r103715 - r103717;
        double r103719 = r103718 * r103703;
        double r103720 = r103719 / r103708;
        double r103721 = r103706 * r103720;
        double r103722 = -3.040807945417772e-242;
        bool r103723 = r103703 <= r103722;
        double r103724 = sqrt(r103706);
        double r103725 = r103708 / r103703;
        double r103726 = r103707 + r103725;
        double r103727 = pow(r103726, r103703);
        double r103728 = r103727 - r103707;
        double r103729 = r103728 / r103725;
        double r103730 = r103724 * r103729;
        double r103731 = r103724 * r103730;
        double r103732 = 1.4887262418381814e-106;
        bool r103733 = r103703 <= r103732;
        double r103734 = 1.0;
        double r103735 = fma(r103712, r103703, r103734);
        double r103736 = fma(r103707, r103708, r103735);
        double r103737 = r103736 - r103707;
        double r103738 = r103737 / r103725;
        double r103739 = r103706 * r103738;
        double r103740 = cbrt(r103718);
        double r103741 = r103740 * r103740;
        double r103742 = r103706 * r103741;
        double r103743 = cbrt(r103703);
        double r103744 = r103743 * r103743;
        double r103745 = r103742 * r103744;
        double r103746 = r103745 / r103708;
        double r103747 = r103734 / r103743;
        double r103748 = r103740 / r103747;
        double r103749 = r103746 * r103748;
        double r103750 = r103733 ? r103739 : r103749;
        double r103751 = r103723 ? r103731 : r103750;
        double r103752 = r103705 ? r103721 : r103751;
        return r103752;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	43.2
Herbie	23.5

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

Derivation

1. Split input into 4 regimes

2. if n < -2.2600474911734306e+95

   1. Initial program 49.3

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

   3. Applied div-inv 49.3

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

   4. Applied *-un-lft-identity 49.3

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

   5. Applied times-frac 48.9

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

   6. Applied associate-*r* 48.9

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

   7. Simplified 48.8

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

   8. Taylor expanded around 0 25.9

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

   9. Simplified 25.9

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\]
   10. Using strategy rm

   11. Applied div-inv 25.9

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

   12. Applied associate-*l* 25.6

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

   13. Simplified 25.4

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

   if -2.2600474911734306e+95 < n < -3.040807945417772e-242

   1. Initial program 24.4

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

   3. Applied add-sqr-sqrt 24.4

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

   4. Applied associate-*l* 24.4

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

   if -3.040807945417772e-242 < n < 1.4887262418381814e-106

   1. Initial program 35.6

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

   2. Taylor expanded around 0 29.6

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

   3. Simplified 29.6

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

   if 1.4887262418381814e-106 < n

   1. Initial program 60.4

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

   3. Applied div-inv 60.4

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

   4. Applied *-un-lft-identity 60.4

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

   5. Applied times-frac 60.1

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

   6. Applied associate-*r* 60.1

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

   7. Simplified 60.1

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

   8. Taylor expanded around 0 18.9

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

   9. Simplified 18.9

      \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{n}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 19.5

       \[\leadsto \frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}\]

   12. Applied *-un-lft-identity 19.5

       \[\leadsto \frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}\]

   13. Applied times-frac 19.5

       \[\leadsto \frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\color{blue}{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{1}{\sqrt[3]{n}}}}\]

   14. Applied add-cube-cbrt 19.8

       \[\leadsto \frac{100}{i} \cdot \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}}{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{1}{\sqrt[3]{n}}}\]

   15. Applied times-frac 19.7

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(\frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{\sqrt[3]{n}}}\right)}\]

   16. Applied associate-*r* 18.1

       \[\leadsto \color{blue}{\left(\frac{100}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{\sqrt[3]{n}}}}\]

   17. Simplified 17.7

       \[\leadsto \color{blue}{\frac{\left(100 \cdot \left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right)\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)}{i}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{\sqrt[3]{n}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 23.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.260047491173430560879176932933542305034 \cdot 10^{95}:\\ \;\;\;\;100 \cdot \frac{\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;n \le -3.040807945417772148751844957499095808404 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.48872624183818144300366844690655908205 \cdot 10^{-106}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(100 \cdot \left(\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right)\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{1}{\sqrt[3]{n}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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))))