Compound Interest

Average Error: 43.1 → 30.0

Time: 31.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -42606.74875587941642152145504951477050781:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(\frac{i}{n}\right)}^{n} - 1}}\\ \mathbf{elif}\;i \le 3.332988420430884258276291203877273900307 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.944860624286458495633005962801713322847 \cdot 10^{141}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{1}{\frac{i}{\left(\mathsf{fma}\left(\frac{1}{2}, {\left(\log i\right)}^{2} \cdot {n}^{2}, \mathsf{fma}\left(\frac{1}{2}, {n}^{2} \cdot {\left(\log n\right)}^{2}, \mathsf{fma}\left(\frac{1}{6}, {\left(\log i\right)}^{3} \cdot {n}^{3}, \mathsf{fma}\left(\log i, n, \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right)\right)\right) - \log n \cdot \mathsf{fma}\left(n \cdot n, \log i, n\right)\right) - \mathsf{fma}\left(\frac{1}{2}, {\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right), \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)}}\\ \mathbf{elif}\;i \le 2.568245662391043108180991997152758915531 \cdot 10^{231}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r167095 = 100.0;
        double r167096 = 1.0;
        double r167097 = i;
        double r167098 = n;
        double r167099 = r167097 / r167098;
        double r167100 = r167096 + r167099;
        double r167101 = pow(r167100, r167098);
        double r167102 = r167101 - r167096;
        double r167103 = r167102 / r167099;
        double r167104 = r167095 * r167103;
        return r167104;
}

↓

double f(double i, double n) {
        double r167105 = i;
        double r167106 = -42606.74875587942;
        bool r167107 = r167105 <= r167106;
        double r167108 = 100.0;
        double r167109 = n;
        double r167110 = r167105 / r167109;
        double r167111 = pow(r167110, r167109);
        double r167112 = 1.0;
        double r167113 = r167111 - r167112;
        double r167114 = r167105 / r167113;
        double r167115 = r167109 / r167114;
        double r167116 = r167108 * r167115;
        double r167117 = 3.332988420430884e-27;
        bool r167118 = r167105 <= r167117;
        double r167119 = 0.5;
        double r167120 = 2.0;
        double r167121 = pow(r167105, r167120);
        double r167122 = log(r167112);
        double r167123 = r167122 * r167109;
        double r167124 = fma(r167119, r167121, r167123);
        double r167125 = fma(r167112, r167105, r167124);
        double r167126 = r167121 * r167122;
        double r167127 = r167119 * r167126;
        double r167128 = r167125 - r167127;
        double r167129 = r167128 / r167110;
        double r167130 = r167108 * r167129;
        double r167131 = 9.944860624286458e+141;
        bool r167132 = r167105 <= r167131;
        double r167133 = r167108 * r167109;
        double r167134 = 1.0;
        double r167135 = 0.5;
        double r167136 = log(r167105);
        double r167137 = pow(r167136, r167120);
        double r167138 = pow(r167109, r167120);
        double r167139 = r167137 * r167138;
        double r167140 = log(r167109);
        double r167141 = pow(r167140, r167120);
        double r167142 = r167138 * r167141;
        double r167143 = 0.16666666666666666;
        double r167144 = 3.0;
        double r167145 = pow(r167136, r167144);
        double r167146 = pow(r167109, r167144);
        double r167147 = r167145 * r167146;
        double r167148 = r167146 * r167141;
        double r167149 = r167136 * r167148;
        double r167150 = r167135 * r167149;
        double r167151 = fma(r167136, r167109, r167150);
        double r167152 = fma(r167143, r167147, r167151);
        double r167153 = fma(r167135, r167142, r167152);
        double r167154 = fma(r167135, r167139, r167153);
        double r167155 = r167109 * r167109;
        double r167156 = fma(r167155, r167136, r167109);
        double r167157 = r167140 * r167156;
        double r167158 = r167154 - r167157;
        double r167159 = r167146 * r167140;
        double r167160 = r167137 * r167159;
        double r167161 = pow(r167140, r167144);
        double r167162 = r167146 * r167161;
        double r167163 = r167143 * r167162;
        double r167164 = fma(r167135, r167160, r167163);
        double r167165 = r167158 - r167164;
        double r167166 = r167105 / r167165;
        double r167167 = r167134 / r167166;
        double r167168 = r167133 * r167167;
        double r167169 = 2.568245662391043e+231;
        bool r167170 = r167105 <= r167169;
        double r167171 = r167112 + r167110;
        double r167172 = r167109 / r167120;
        double r167173 = pow(r167171, r167172);
        double r167174 = sqrt(r167112);
        double r167175 = r167173 + r167174;
        double r167176 = r167175 / r167105;
        double r167177 = r167173 - r167174;
        double r167178 = r167177 * r167109;
        double r167179 = r167176 * r167178;
        double r167180 = r167108 * r167179;
        double r167181 = fma(r167122, r167109, r167134);
        double r167182 = fma(r167112, r167105, r167181);
        double r167183 = r167182 - r167112;
        double r167184 = r167183 / r167110;
        double r167185 = r167108 * r167184;
        double r167186 = r167170 ? r167180 : r167185;
        double r167187 = r167132 ? r167168 : r167186;
        double r167188 = r167118 ? r167130 : r167187;
        double r167189 = r167107 ? r167116 : r167188;
        return r167189;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.8
Herbie	30.0

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

Derivation

1. Split input into 5 regimes

2. if i < -42606.74875587942

   1. Initial program 27.8

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 18.8

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

   if -42606.74875587942 < i < 3.332988420430884e-27

   1. Initial program 50.7

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

   2. Taylor expanded around 0 34.2

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

   3. Simplified 34.2

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

   if 3.332988420430884e-27 < i < 9.944860624286458e+141

   1. Initial program 38.7

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

   2. Taylor expanded around inf 37.2

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

   3. Simplified 39.2

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

   5. Applied div-inv 39.2

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

   6. Applied associate-*r* 39.2

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

   7. Taylor expanded around 0 22.8

      \[\leadsto \left(100 \cdot n\right) \cdot \frac{1}{\frac{i}{\color{blue}{\left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2}\right) + \left(\frac{1}{2} \cdot \left({n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \left(\log i \cdot n + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right)\right)\right) - \left(n \cdot \log n + \left(\log i \cdot \left({n}^{2} \cdot \log n\right) + \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)}}}\]

   8. Simplified 22.8

      \[\leadsto \left(100 \cdot n\right) \cdot \frac{1}{\frac{i}{\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, {\left(\log i\right)}^{2} \cdot {n}^{2}, \mathsf{fma}\left(\frac{1}{2}, {n}^{2} \cdot {\left(\log n\right)}^{2}, \mathsf{fma}\left(\frac{1}{6}, {\left(\log i\right)}^{3} \cdot {n}^{3}, \mathsf{fma}\left(\log i, n, \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right)\right)\right) - \log n \cdot \mathsf{fma}\left(n \cdot n, \log i, n\right)\right) - \mathsf{fma}\left(\frac{1}{2}, {\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right), \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)}}}\]

   if 9.944860624286458e+141 < i < 2.568245662391043e+231

   1. Initial program 32.0

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

   3. Applied div-inv 32.0

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

   4. Applied add-sqr-sqrt 32.0

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

   5. Applied sqr-pow 32.1

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

   6. Applied difference-of-squares 32.0

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

   7. Applied times-frac 32.0

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

   8. Simplified 32.0

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

   if 2.568245662391043e+231 < i

   1. Initial program 29.1

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

   2. Taylor expanded around 0 36.1

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

   3. Simplified 36.1

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

4. Final simplification 30.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -42606.74875587941642152145504951477050781:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(\frac{i}{n}\right)}^{n} - 1}}\\ \mathbf{elif}\;i \le 3.332988420430884258276291203877273900307 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.944860624286458495633005962801713322847 \cdot 10^{141}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{1}{\frac{i}{\left(\mathsf{fma}\left(\frac{1}{2}, {\left(\log i\right)}^{2} \cdot {n}^{2}, \mathsf{fma}\left(\frac{1}{2}, {n}^{2} \cdot {\left(\log n\right)}^{2}, \mathsf{fma}\left(\frac{1}{6}, {\left(\log i\right)}^{3} \cdot {n}^{3}, \mathsf{fma}\left(\log i, n, \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right)\right)\right) - \log n \cdot \mathsf{fma}\left(n \cdot n, \log i, n\right)\right) - \mathsf{fma}\left(\frac{1}{2}, {\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right), \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)}}\\ \mathbf{elif}\;i \le 2.568245662391043108180991997152758915531 \cdot 10^{231}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} + \sqrt{1}}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} - \sqrt{1}\right) \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))