Compound Interest

Average Error: 43.2 → 21.6

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 5.88981601408302989 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \left(\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot \mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{2}, \log 1 \cdot n, 1\right) - \sqrt{1}\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r178403 = 100.0;
        double r178404 = 1.0;
        double r178405 = i;
        double r178406 = n;
        double r178407 = r178405 / r178406;
        double r178408 = r178404 + r178407;
        double r178409 = pow(r178408, r178406);
        double r178410 = r178409 - r178404;
        double r178411 = r178410 / r178407;
        double r178412 = r178403 * r178411;
        return r178412;
}

↓

double f(double i, double n) {
        double r178413 = i;
        double r178414 = -6.8977356701701615e-09;
        bool r178415 = r178413 <= r178414;
        double r178416 = 100.0;
        double r178417 = 1.0;
        double r178418 = n;
        double r178419 = r178413 / r178418;
        double r178420 = r178417 + r178419;
        double r178421 = pow(r178420, r178418);
        double r178422 = r178421 / r178419;
        double r178423 = r178417 / r178419;
        double r178424 = r178422 - r178423;
        double r178425 = r178416 * r178424;
        double r178426 = 5.88981601408303e-08;
        bool r178427 = r178413 <= r178426;
        double r178428 = 2.0;
        double r178429 = r178418 / r178428;
        double r178430 = pow(r178420, r178429);
        double r178431 = sqrt(r178417);
        double r178432 = r178430 + r178431;
        double r178433 = 0.5;
        double r178434 = 0.5;
        double r178435 = log(r178417);
        double r178436 = r178435 * r178418;
        double r178437 = 1.0;
        double r178438 = fma(r178434, r178436, r178437);
        double r178439 = r178438 - r178431;
        double r178440 = fma(r178413, r178433, r178439);
        double r178441 = r178432 * r178440;
        double r178442 = r178441 / r178413;
        double r178443 = r178442 * r178418;
        double r178444 = r178416 * r178443;
        double r178445 = r178432 * r178416;
        double r178446 = r178430 - r178431;
        double r178447 = r178446 / r178419;
        double r178448 = r178445 * r178447;
        double r178449 = r178427 ? r178444 : r178448;
        double r178450 = r178415 ? r178425 : r178449;
        return r178450;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.2
Target	42.9
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 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 < 5.88981601408303e-08

   1. Initial program 51.2

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

   3. Applied div-inv 51.2

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

   4. Applied add-sqr-sqrt 51.2

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

   5. Applied sqr-pow 51.2

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

   6. Applied difference-of-squares 51.2

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

   7. Applied times-frac 50.9

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

   8. Simplified 50.9

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

   9. Taylor expanded around 0 50.7

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

   10. Simplified 15.5

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

   12. Applied associate-*r* 16.3

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

   14. Applied associate-*l/ 16.2

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

   if 5.88981601408303e-08 < 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 21.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -6.8977356701701615 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 5.88981601408302989 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \left(\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot \mathsf{fma}\left(i, 0.5, \mathsf{fma}\left(\frac{1}{2}, \log 1 \cdot n, 1\right) - \sqrt{1}\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}\right) \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}}{\frac{i}{n}}\\ \end{array}\]

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))))