Compound Interest

Average Error: 47.8 → 16.9

Time: 13.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 3.9028797563160959 \cdot 10^{-7}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r158661 = 100.0;
        double r158662 = 1.0;
        double r158663 = i;
        double r158664 = n;
        double r158665 = r158663 / r158664;
        double r158666 = r158662 + r158665;
        double r158667 = pow(r158666, r158664);
        double r158668 = r158667 - r158662;
        double r158669 = r158668 / r158665;
        double r158670 = r158661 * r158669;
        return r158670;
}

↓

double f(double i, double n) {
        double r158671 = i;
        double r158672 = -6.8977356701701615e-09;
        bool r158673 = r158671 <= r158672;
        double r158674 = 100.0;
        double r158675 = 1.0;
        double r158676 = n;
        double r158677 = r158671 / r158676;
        double r158678 = r158675 + r158677;
        double r158679 = pow(r158678, r158676);
        double r158680 = r158679 / r158677;
        double r158681 = r158675 / r158677;
        double r158682 = r158680 - r158681;
        double r158683 = r158674 * r158682;
        double r158684 = 3.902879756316096e-07;
        bool r158685 = r158671 <= r158684;
        double r158686 = 0.5;
        double r158687 = 2.0;
        double r158688 = pow(r158671, r158687);
        double r158689 = log(r158675);
        double r158690 = r158689 * r158676;
        double r158691 = fma(r158686, r158688, r158690);
        double r158692 = r158688 * r158689;
        double r158693 = r158686 * r158692;
        double r158694 = r158691 - r158693;
        double r158695 = fma(r158671, r158675, r158694);
        double r158696 = r158695 / r158671;
        double r158697 = r158674 * r158696;
        double r158698 = r158697 * r158676;
        double r158699 = r158676 / r158687;
        double r158700 = pow(r158678, r158699);
        double r158701 = sqrt(r158675);
        double r158702 = r158700 + r158701;
        double r158703 = r158702 * r158674;
        double r158704 = r158700 - r158701;
        double r158705 = r158704 / r158677;
        double r158706 = r158703 * r158705;
        double r158707 = r158685 ? r158698 : r158706;
        double r158708 = r158673 ? r158683 : r158707;
        return r158708;
}

Error

Bits error versus i

Bits error versus n

Target

Original	47.8
Target	47.4
Herbie	16.9

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

Derivation

1. Split input into 3 regimes

2. if i < -6.8977356701701615e-09

   1. Initial program 29.3

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

   3. Applied div-sub 29.3

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

   if -6.8977356701701615e-09 < i < 3.902879756316096e-07

   1. Initial program 58.5

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

   2. Taylor expanded around 0 26.3

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

   3. Simplified 26.3

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

   5. Applied associate-/r/ 8.6

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

   6. Applied associate-*r* 8.6

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

   if 3.902879756316096e-07 < i

   1. Initial program 32.1

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

   3. Applied *-un-lft-identity 32.1

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

   4. Applied *-un-lft-identity 32.1

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

   5. Applied times-frac 32.1

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

   6. Applied add-sqr-sqrt 32.1

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

   7. Applied sqr-pow 32.2

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

   8. Applied difference-of-squares 32.2

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

   9. Applied times-frac 32.2

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

   10. Applied associate-*r* 32.2

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

   11. Simplified 32.2

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

4. Final simplification 16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 3.9028797563160959 \cdot 10^{-7}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))