Average Error: 42.6 → 21.6
Time: 38.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\ \mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\
\;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\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)} \cdot \sqrt[3]{\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)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\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)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\

\mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\
\;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{i}}{\frac{1}{n}}\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r217149 = 100.0;
        double r217150 = 1.0;
        double r217151 = i;
        double r217152 = n;
        double r217153 = r217151 / r217152;
        double r217154 = r217150 + r217153;
        double r217155 = pow(r217154, r217152);
        double r217156 = r217155 - r217150;
        double r217157 = r217156 / r217153;
        double r217158 = r217149 * r217157;
        return r217158;
}
↓
double f(double i, double n) {
        double r217159 = i;
        double r217160 = -1.7771586426478155e-06;
        bool r217161 = r217159 <= r217160;
        double r217162 = 100.0;
        double r217163 = 1.0;
        double r217164 = n;
        double r217165 = r217159 / r217164;
        double r217166 = r217163 + r217165;
        double r217167 = 2.0;
        double r217168 = r217167 * r217164;
        double r217169 = pow(r217166, r217168);
        double r217170 = r217163 * r217163;
        double r217171 = r217169 - r217170;
        double r217172 = pow(r217166, r217164);
        double r217173 = r217172 + r217163;
        double r217174 = r217171 / r217173;
        double r217175 = r217174 / r217165;
        double r217176 = r217162 * r217175;
        double r217177 = 2.882168870282429e-18;
        bool r217178 = r217159 <= r217177;
        double r217179 = 1.0;
        double r217180 = cbrt(r217159);
        double r217181 = r217180 * r217180;
        double r217182 = r217179 / r217181;
        double r217183 = r217179 / r217164;
        double r217184 = r217163 * r217159;
        double r217185 = 0.5;
        double r217186 = pow(r217159, r217167);
        double r217187 = r217185 * r217186;
        double r217188 = log(r217163);
        double r217189 = r217188 * r217164;
        double r217190 = r217187 + r217189;
        double r217191 = r217184 + r217190;
        double r217192 = r217186 * r217188;
        double r217193 = r217185 * r217192;
        double r217194 = r217191 - r217193;
        double r217195 = cbrt(r217194);
        double r217196 = r217195 * r217195;
        double r217197 = cbrt(r217181);
        double r217198 = r217196 / r217197;
        double r217199 = cbrt(r217180);
        double r217200 = r217195 / r217199;
        double r217201 = r217198 * r217200;
        double r217202 = r217183 / r217201;
        double r217203 = r217182 / r217202;
        double r217204 = r217162 * r217203;
        double r217205 = 2.1606840039657686e+162;
        bool r217206 = r217159 <= r217205;
        double r217207 = 3.0;
        double r217208 = pow(r217172, r217207);
        double r217209 = pow(r217163, r217207);
        double r217210 = r217208 - r217209;
        double r217211 = r217163 * r217173;
        double r217212 = r217169 + r217211;
        double r217213 = r217210 / r217212;
        double r217214 = r217213 / r217159;
        double r217215 = r217214 / r217183;
        double r217216 = r217162 * r217215;
        double r217217 = r217189 + r217179;
        double r217218 = r217184 + r217217;
        double r217219 = r217218 - r217163;
        double r217220 = r217219 / r217165;
        double r217221 = r217162 * r217220;
        double r217222 = r217206 ? r217216 : r217221;
        double r217223 = r217178 ? r217204 : r217222;
        double r217224 = r217161 ? r217176 : r217223;
        return r217224;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.6
Target42.5
Herbie21.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.7771586426478155e-06

    1. Initial program 27.7

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

    3. Applied flip--27.7

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

    4. Simplified27.7

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

    if -1.7771586426478155e-06 < i < 2.882168870282429e-18

    1. Initial program 50.5

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

    3. Applied div-inv50.5

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

    4. Applied associate-/r*50.2

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

    5. Taylor expanded around 0 16.7

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

    7. Applied add-cube-cbrt17.6

      \[\leadsto 100 \cdot \frac{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\color{blue}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}}{\frac{1}{n}}\]

    8. Applied *-un-lft-identity17.6

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{1 \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}{\frac{1}{n}}\]

    9. Applied times-frac17.6

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{i}}}}{\frac{1}{n}}\]

    10. Applied associate-/l*16.1

      \[\leadsto 100 \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{i}}}}}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt16.2

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}}}}\]

    13. Applied cbrt-prod16.3

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\color{blue}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \sqrt[3]{\sqrt[3]{i}}}}}}\]

    14. Applied add-cube-cbrt15.8

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\color{blue}{\left(\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right) \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \sqrt[3]{\sqrt[3]{i}}}}}\]

    15. Applied times-frac15.8

      \[\leadsto 100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\color{blue}{\frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}}\]

    if 2.882168870282429e-18 < i < 2.1606840039657686e+162

    1. Initial program 36.1

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

    3. Applied div-inv36.1

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

    4. Applied associate-/r*36.1

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

    6. Applied flip3--36.1

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

    7. Simplified36.1

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

    if 2.1606840039657686e+162 < i

    1. Initial program 31.9

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

    2. Taylor expanded around 0 35.5

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification21.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\ \mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right)}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))