Compound Interest

Average Error: 42.2 → 24.2

Time: 35.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -2.260823056891884243812192182030091462802 \cdot 10^{130}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.971792173734567753657378996256742049344 \cdot 10^{92}:\\ \;\;\;\;\left(\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right) \cdot n\\ \mathbf{elif}\;n \le -3.460770537187881137122290954175863408168 \cdot 10^{79} \lor \neg \left(n \le 6.434758453518077622758166934880438333831 \cdot 10^{-141}\right):\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r244542 = 100.0;
        double r244543 = 1.0;
        double r244544 = i;
        double r244545 = n;
        double r244546 = r244544 / r244545;
        double r244547 = r244543 + r244546;
        double r244548 = pow(r244547, r244545);
        double r244549 = r244548 - r244543;
        double r244550 = r244549 / r244546;
        double r244551 = r244542 * r244550;
        return r244551;
}

↓

double f(double i, double n) {
        double r244552 = n;
        double r244553 = -2.260823056891884e+130;
        bool r244554 = r244552 <= r244553;
        double r244555 = 100.0;
        double r244556 = 1.0;
        double r244557 = i;
        double r244558 = log(r244556);
        double r244559 = r244557 * r244557;
        double r244560 = 0.5;
        double r244561 = r244560 * r244558;
        double r244562 = r244560 - r244561;
        double r244563 = r244559 * r244562;
        double r244564 = fma(r244552, r244558, r244563);
        double r244565 = fma(r244556, r244557, r244564);
        double r244566 = r244565 / r244557;
        double r244567 = r244555 * r244566;
        double r244568 = r244567 * r244552;
        double r244569 = -4.971792173734568e+92;
        bool r244570 = r244552 <= r244569;
        double r244571 = sqrt(r244555);
        double r244572 = r244557 / r244552;
        double r244573 = r244556 + r244572;
        double r244574 = pow(r244573, r244552);
        double r244575 = r244574 - r244556;
        double r244576 = r244575 / r244557;
        double r244577 = r244571 * r244576;
        double r244578 = r244571 * r244577;
        double r244579 = r244578 * r244552;
        double r244580 = -3.460770537187881e+79;
        bool r244581 = r244552 <= r244580;
        double r244582 = 6.434758453518078e-141;
        bool r244583 = r244552 <= r244582;
        double r244584 = !r244583;
        bool r244585 = r244581 || r244584;
        double r244586 = exp(r244575);
        double r244587 = log(r244586);
        double r244588 = r244587 / r244572;
        double r244589 = r244555 * r244588;
        double r244590 = r244585 ? r244568 : r244589;
        double r244591 = r244570 ? r244579 : r244590;
        double r244592 = r244554 ? r244568 : r244591;
        return r244592;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.2
Target	42.3
Herbie	24.2

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

Derivation

1. Split input into 3 regimes

2. if n < -2.260823056891884e+130 or -4.971792173734568e+92 < n < -3.460770537187881e+79 or 6.434758453518078e-141 < n

   1. Initial program 55.8

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

   3. Applied associate-/r/ 55.4

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

   4. Applied associate-*r* 55.4

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

   5. Taylor expanded around 0 21.1

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

   6. Simplified 21.1

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

   if -2.260823056891884e+130 < n < -4.971792173734568e+92

   1. Initial program 38.6

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

   3. Applied associate-/r/ 38.4

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

   4. Applied associate-*r* 38.3

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

   6. Applied add-sqr-sqrt 38.3

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

   7. Applied associate-*l* 38.4

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

   if -3.460770537187881e+79 < n < 6.434758453518078e-141

   1. Initial program 26.5

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

   3. Applied add-log-exp 26.5

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

   4. Applied add-log-exp 26.5

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

   5. Applied diff-log 26.5

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

   6. Simplified 26.5

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

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.260823056891884243812192182030091462802 \cdot 10^{130}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.971792173734567753657378996256742049344 \cdot 10^{92}:\\ \;\;\;\;\left(\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right) \cdot n\\ \mathbf{elif}\;n \le -3.460770537187881137122290954175863408168 \cdot 10^{79} \lor \neg \left(n \le 6.434758453518077622758166934880438333831 \cdot 10^{-141}\right):\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(n, \log 1, \left(i \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \log 1\right)\right)\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))))