Compound Interest

Average Error: 43.4 → 20.5

Time: 23.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(\mathsf{fma}\left(n, \log 1, i \cdot \left(0.5 \cdot i + 1\right)\right) - \left(0.5 \cdot \left(i \cdot i\right)\right) \cdot \log 1\right)\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i \cdot \mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)}{n}}\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\\ \end{array}\]

C

double f(double i, double n) {
        double r102475 = 100.0;
        double r102476 = 1.0;
        double r102477 = i;
        double r102478 = n;
        double r102479 = r102477 / r102478;
        double r102480 = r102476 + r102479;
        double r102481 = pow(r102480, r102478);
        double r102482 = r102481 - r102476;
        double r102483 = r102482 / r102479;
        double r102484 = r102475 * r102483;
        return r102484;
}

↓

double f(double i, double n) {
        double r102485 = i;
        double r102486 = -0.9893212200252494;
        bool r102487 = r102485 <= r102486;
        double r102488 = 100.0;
        double r102489 = n;
        double r102490 = r102485 / r102489;
        double r102491 = 1.0;
        double r102492 = r102490 + r102491;
        double r102493 = 2.0;
        double r102494 = r102489 * r102493;
        double r102495 = pow(r102492, r102494);
        double r102496 = r102491 * r102491;
        double r102497 = r102495 - r102496;
        double r102498 = pow(r102492, r102489);
        double r102499 = r102498 + r102491;
        double r102500 = r102497 / r102499;
        double r102501 = r102500 / r102490;
        double r102502 = r102488 * r102501;
        double r102503 = 1.5086524363033694e-09;
        bool r102504 = r102485 <= r102503;
        double r102505 = log(r102491);
        double r102506 = 0.5;
        double r102507 = r102506 * r102485;
        double r102508 = r102507 + r102491;
        double r102509 = r102485 * r102508;
        double r102510 = fma(r102489, r102505, r102509);
        double r102511 = r102485 * r102485;
        double r102512 = r102506 * r102511;
        double r102513 = r102512 * r102505;
        double r102514 = r102510 - r102513;
        double r102515 = r102489 * r102514;
        double r102516 = 1.0;
        double r102517 = r102516 / r102485;
        double r102518 = r102515 * r102517;
        double r102519 = r102488 * r102518;
        double r102520 = 8.608246042115979e+235;
        bool r102521 = r102485 <= r102520;
        double r102522 = 3.0;
        double r102523 = pow(r102498, r102522);
        double r102524 = pow(r102491, r102522);
        double r102525 = r102523 - r102524;
        double r102526 = fma(r102491, r102499, r102495);
        double r102527 = r102485 * r102526;
        double r102528 = r102527 / r102489;
        double r102529 = r102525 / r102528;
        double r102530 = r102488 * r102529;
        double r102531 = 1.844738975002478e+296;
        bool r102532 = r102485 <= r102531;
        double r102533 = fma(r102485, r102491, r102516);
        double r102534 = fma(r102489, r102505, r102533);
        double r102535 = r102534 - r102491;
        double r102536 = r102535 / r102490;
        double r102537 = r102536 * r102488;
        double r102538 = r102488 * r102489;
        double r102539 = r102498 - r102491;
        double r102540 = r102539 / r102485;
        double r102541 = r102538 * r102540;
        double r102542 = r102532 ? r102537 : r102541;
        double r102543 = r102521 ? r102530 : r102542;
        double r102544 = r102504 ? r102519 : r102543;
        double r102545 = r102487 ? r102502 : r102544;
        return r102545;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.4
Target	43.3
Herbie	20.5

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

Derivation

1. Split input into 5 regimes

2. if i < -0.9893212200252494

   1. Initial program 27.9

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

   3. Applied flip-- 27.9

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

   4. Simplified 27.9

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

   5. Simplified 27.9

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

   if -0.9893212200252494 < i < 1.5086524363033694e-09

   1. Initial program 51.2

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

   2. Taylor expanded around 0 32.6

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

   3. Simplified 32.6

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

   5. Applied div-inv 32.7

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

   6. Applied *-un-lft-identity 32.7

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

   7. Applied times-frac 15.1

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

   8. Simplified 15.1

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

   if 1.5086524363033694e-09 < i < 8.608246042115979e+235

   1. Initial program 33.3

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

   3. Applied flip3-- 33.3

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

   4. Applied associate-/l/ 33.3

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

   5. Simplified 33.3

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

   if 8.608246042115979e+235 < i < 1.844738975002478e+296

   1. Initial program 32.1

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

   2. Taylor expanded around 0 34.0

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

   3. Simplified 34.0

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

   if 1.844738975002478e+296 < i

   1. Initial program 28.4

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

   3. Applied pow1 28.4

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

   4. Applied pow1 28.4

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

   5. Applied pow-prod-down 28.4

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

   6. Simplified 28.4

      \[\leadsto {\color{blue}{\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)\right)}}^{1}\]
3. Recombined 5 regimes into one program.

4. Final simplification 20.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.9893212200252493593310987307631876319647:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.508652436303369411416630768794228112384 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \left(\mathsf{fma}\left(n, \log 1, i \cdot \left(0.5 \cdot i + 1\right)\right) - \left(0.5 \cdot \left(i \cdot i\right)\right) \cdot \log 1\right)\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;i \le 8.608246042115979009006453659316669705257 \cdot 10^{235}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i \cdot \mathsf{fma}\left(1, {\left(\frac{i}{n} + 1\right)}^{n} + 1, {\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)}\right)}{n}}\\ \mathbf{elif}\;i \le 1.844738975002478002763642496553644874398 \cdot 10^{296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))))