Average Error: 42.7 → 33.3
Time: 21.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -7.3291384716741002 \cdot 10^{181}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -9.8037587964115943 \cdot 10^{-219}:\\ \;\;\;\;100 \cdot \left(\left(\sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}}\right) \cdot \sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}}\right)\\ \mathbf{elif}\;n \le 1.54725910109650585 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -7.3291384716741002 \cdot 10^{181}:\\
\;\;\;\;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)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -9.8037587964115943 \cdot 10^{-219}:\\
\;\;\;\;100 \cdot \left(\left(\sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}}\right) \cdot \sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}}\right)\\

\mathbf{elif}\;n \le 1.54725910109650585 \cdot 10^{-128}:\\
\;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\right)\\

\mathbf{else}:\\
\;\;\;\;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)}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r180548 = 100.0;
        double r180549 = 1.0;
        double r180550 = i;
        double r180551 = n;
        double r180552 = r180550 / r180551;
        double r180553 = r180549 + r180552;
        double r180554 = pow(r180553, r180551);
        double r180555 = r180554 - r180549;
        double r180556 = r180555 / r180552;
        double r180557 = r180548 * r180556;
        return r180557;
}
↓
double f(double i, double n) {
        double r180558 = n;
        double r180559 = -7.3291384716741e+181;
        bool r180560 = r180558 <= r180559;
        double r180561 = 100.0;
        double r180562 = i;
        double r180563 = 1.0;
        double r180564 = 0.5;
        double r180565 = 2.0;
        double r180566 = pow(r180562, r180565);
        double r180567 = log(r180563);
        double r180568 = r180567 * r180558;
        double r180569 = fma(r180564, r180566, r180568);
        double r180570 = r180566 * r180567;
        double r180571 = r180564 * r180570;
        double r180572 = r180569 - r180571;
        double r180573 = fma(r180562, r180563, r180572);
        double r180574 = r180562 / r180558;
        double r180575 = r180573 / r180574;
        double r180576 = r180561 * r180575;
        double r180577 = -9.803758796411594e-219;
        bool r180578 = r180558 <= r180577;
        double r180579 = r180563 + r180574;
        double r180580 = cbrt(r180579);
        double r180581 = r180580 * r180580;
        double r180582 = r180565 * r180558;
        double r180583 = pow(r180581, r180582);
        double r180584 = pow(r180580, r180582);
        double r180585 = r180563 * r180563;
        double r180586 = -r180585;
        double r180587 = fma(r180583, r180584, r180586);
        double r180588 = pow(r180579, r180558);
        double r180589 = r180588 + r180563;
        double r180590 = r180587 / r180589;
        double r180591 = r180590 / r180574;
        double r180592 = cbrt(r180591);
        double r180593 = r180592 * r180592;
        double r180594 = r180593 * r180592;
        double r180595 = r180561 * r180594;
        double r180596 = 1.5472591010965058e-128;
        bool r180597 = r180558 <= r180596;
        double r180598 = sqrt(r180561);
        double r180599 = 1.0;
        double r180600 = fma(r180567, r180558, r180599);
        double r180601 = fma(r180563, r180562, r180600);
        double r180602 = r180601 - r180563;
        double r180603 = r180602 / r180574;
        double r180604 = r180598 * r180603;
        double r180605 = r180598 * r180604;
        double r180606 = r180597 ? r180605 : r180576;
        double r180607 = r180578 ? r180595 : r180606;
        double r180608 = r180560 ? r180576 : r180607;
        return r180608;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.3
Herbie33.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -7.3291384716741e+181 or 1.5472591010965058e-128 < n

    1. Initial program 58.4

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

    2. Taylor expanded around 0 39.8

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

    3. Simplified39.8

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

    if -7.3291384716741e+181 < n < -9.803758796411594e-219

    1. Initial program 28.1

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

    3. Applied flip--28.1

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

    4. Simplified28.0

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

    6. Applied add-cube-cbrt28.1

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

    7. Applied unpow-prod-down28.1

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

    8. Applied fma-def28.1

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

    10. Applied add-cube-cbrt28.3

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

    if -9.803758796411594e-219 < n < 1.5472591010965058e-128

    1. Initial program 32.9

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

    3. Applied add-sqr-sqrt32.9

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

    4. Applied associate-*l*32.9

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

    5. Taylor expanded around 0 27.2

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

    6. Simplified27.2

      \[\leadsto \sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification33.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -7.3291384716741002 \cdot 10^{181}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -9.8037587964115943 \cdot 10^{-219}:\\ \;\;\;\;100 \cdot \left(\left(\sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}} \cdot \sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}}\right) \cdot \sqrt[3]{\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{1 + \frac{i}{n}} \cdot \sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, {\left(\sqrt[3]{1 + \frac{i}{n}}\right)}^{\left(2 \cdot n\right)}, -1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}}\right)\\ \mathbf{elif}\;n \le 1.54725910109650585 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))))