Compound Interest

Average Error: 43.2 → 32.5

Time: 55.9s

Precision: 64

Math

\[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.260760569041402 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}{i} \cdot 100.0\right) \cdot n\\ \mathbf{elif}\;n \le -1.999963205744506:\\ \;\;\;\;100.0 \cdot \frac{\log 1.0 \cdot n + \left(\left(1.0 \cdot i + \left(i \cdot i\right) \cdot 0.5\right) - \log 1.0 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8097259091280393 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)}\right) \cdot 100.0\\ \mathbf{elif}\;n \le 7.680781589380277 \cdot 10^{-180}:\\ \;\;\;\;\frac{\left(\left(1.0 \cdot i + 1\right) + \log 1.0 \cdot n\right) - 1.0}{\frac{i}{n}} \cdot 100.0\\ \mathbf{else}:\\ \;\;\;\;100.0 \cdot \frac{\log 1.0 \cdot n + \left(\left(1.0 \cdot i + \left(i \cdot i\right) \cdot 0.5\right) - \log 1.0 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r6331527 = 100.0;
        double r6331528 = 1.0;
        double r6331529 = i;
        double r6331530 = n;
        double r6331531 = r6331529 / r6331530;
        double r6331532 = r6331528 + r6331531;
        double r6331533 = pow(r6331532, r6331530);
        double r6331534 = r6331533 - r6331528;
        double r6331535 = r6331534 / r6331531;
        double r6331536 = r6331527 * r6331535;
        return r6331536;
}

↓

double f(double i, double n) {
        double r6331537 = n;
        double r6331538 = -1.260760569041402e+85;
        bool r6331539 = r6331537 <= r6331538;
        double r6331540 = i;
        double r6331541 = r6331540 / r6331537;
        double r6331542 = 1.0;
        double r6331543 = r6331541 + r6331542;
        double r6331544 = pow(r6331543, r6331537);
        double r6331545 = r6331544 - r6331542;
        double r6331546 = r6331545 / r6331540;
        double r6331547 = 100.0;
        double r6331548 = r6331546 * r6331547;
        double r6331549 = r6331548 * r6331537;
        double r6331550 = -1.999963205744506;
        bool r6331551 = r6331537 <= r6331550;
        double r6331552 = log(r6331542);
        double r6331553 = r6331552 * r6331537;
        double r6331554 = r6331542 * r6331540;
        double r6331555 = r6331540 * r6331540;
        double r6331556 = 0.5;
        double r6331557 = r6331555 * r6331556;
        double r6331558 = r6331554 + r6331557;
        double r6331559 = r6331552 * r6331557;
        double r6331560 = r6331558 - r6331559;
        double r6331561 = r6331553 + r6331560;
        double r6331562 = r6331561 / r6331541;
        double r6331563 = r6331547 * r6331562;
        double r6331564 = -1.8097259091280393e-258;
        bool r6331565 = r6331537 <= r6331564;
        double r6331566 = exp(r6331545);
        double r6331567 = log(r6331566);
        double r6331568 = cbrt(r6331567);
        double r6331569 = cbrt(r6331541);
        double r6331570 = cbrt(r6331569);
        double r6331571 = r6331570 * r6331570;
        double r6331572 = r6331571 * r6331570;
        double r6331573 = r6331568 / r6331572;
        double r6331574 = r6331568 * r6331568;
        double r6331575 = r6331569 * r6331572;
        double r6331576 = r6331574 / r6331575;
        double r6331577 = r6331573 * r6331576;
        double r6331578 = r6331577 * r6331547;
        double r6331579 = 7.680781589380277e-180;
        bool r6331580 = r6331537 <= r6331579;
        double r6331581 = 1.0;
        double r6331582 = r6331554 + r6331581;
        double r6331583 = r6331582 + r6331553;
        double r6331584 = r6331583 - r6331542;
        double r6331585 = r6331584 / r6331541;
        double r6331586 = r6331585 * r6331547;
        double r6331587 = r6331580 ? r6331586 : r6331563;
        double r6331588 = r6331565 ? r6331578 : r6331587;
        double r6331589 = r6331551 ? r6331563 : r6331588;
        double r6331590 = r6331539 ? r6331549 : r6331589;
        return r6331590;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.2
Target	42.9
Herbie	32.5

\[100.0 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1.0 + \frac{i}{n} = 1.0:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1.0 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1.0\right) - 1.0}\\ \end{array}} - 1.0}{\frac{i}{n}}\]

Derivation

1. Split input into 4 regimes

2. if n < -1.260760569041402e+85

   1. Initial program 47.8

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied associate-/r/ 47.4

      \[\leadsto 100.0 \cdot \color{blue}{\left(\frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{i} \cdot n\right)}\]

   4. Applied associate-*r* 47.4

      \[\leadsto \color{blue}{\left(100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{i}\right) \cdot n}\]

   if -1.260760569041402e+85 < n < -1.999963205744506 or 7.680781589380277e-180 < n

   1. Initial program 54.5

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]

   2. Taylor expanded around 0 34.1

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\left(\log 1.0 \cdot n + \left(1.0 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1.0 \cdot {i}^{2}\right)}}{\frac{i}{n}}\]

   3. Simplified 34.1

      \[\leadsto 100.0 \cdot \frac{\color{blue}{n \cdot \log 1.0 + \left(\left(1.0 \cdot i + \left(i \cdot i\right) \cdot 0.5\right) - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1.0\right)}}{\frac{i}{n}}\]

   if -1.999963205744506 < n < -1.8097259091280393e-258

   1. Initial program 17.4

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied add-log-exp 17.4

      \[\leadsto 100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - \color{blue}{\log \left(e^{1.0}\right)}}{\frac{i}{n}}\]

   4. Applied add-log-exp 17.4

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\log \left(e^{{\left(1.0 + \frac{i}{n}\right)}^{n}}\right)} - \log \left(e^{1.0}\right)}{\frac{i}{n}}\]

   5. Applied diff-log 17.4

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\log \left(\frac{e^{{\left(1.0 + \frac{i}{n}\right)}^{n}}}{e^{1.0}}\right)}}{\frac{i}{n}}\]

   6. Simplified 17.4

      \[\leadsto 100.0 \cdot \frac{\log \color{blue}{\left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\frac{i}{n}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 17.4

      \[\leadsto 100.0 \cdot \frac{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}{\color{blue}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}}\]

   9. Applied add-cube-cbrt 17.4

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\left(\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}\]

   10. Applied times-frac 17.4

       \[\leadsto 100.0 \cdot \color{blue}{\left(\frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}}}\right)}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 17.4

       \[\leadsto 100.0 \cdot \left(\frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}}}\right)\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 17.4

       \[\leadsto 100.0 \cdot \left(\frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}}\right)\]

   if -1.8097259091280393e-258 < n < 7.680781589380277e-180

   1. Initial program 30.8

      \[100.0 \cdot \frac{{\left(1.0 + \frac{i}{n}\right)}^{n} - 1.0}{\frac{i}{n}}\]

   2. Taylor expanded around 0 20.6

      \[\leadsto 100.0 \cdot \frac{\color{blue}{\left(\log 1.0 \cdot n + \left(1.0 \cdot i + 1\right)\right)} - 1.0}{\frac{i}{n}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 32.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.260760569041402 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}{i} \cdot 100.0\right) \cdot n\\ \mathbf{elif}\;n \le -1.999963205744506:\\ \;\;\;\;100.0 \cdot \frac{\log 1.0 \cdot n + \left(\left(1.0 \cdot i + \left(i \cdot i\right) \cdot 0.5\right) - \log 1.0 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8097259091280393 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}} \cdot \frac{\sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(\frac{i}{n} + 1.0\right)}^{n} - 1.0}\right)}}{\sqrt[3]{\frac{i}{n}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{i}{n}}}\right)}\right) \cdot 100.0\\ \mathbf{elif}\;n \le 7.680781589380277 \cdot 10^{-180}:\\ \;\;\;\;\frac{\left(\left(1.0 \cdot i + 1\right) + \log 1.0 \cdot n\right) - 1.0}{\frac{i}{n}} \cdot 100.0\\ \mathbf{else}:\\ \;\;\;\;100.0 \cdot \frac{\log 1.0 \cdot n + \left(\left(1.0 \cdot i + \left(i \cdot i\right) \cdot 0.5\right) - \log 1.0 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))))