Compound Interest

Average Error: 43.0 → 25.2

Time: 30.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.9035517225708853 \cdot 10^{-05}:\\ \;\;\;\;\frac{1}{i} \cdot \left(\mathsf{fma}\left(\left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), 100, -100\right) \cdot n\right)\\ \mathbf{elif}\;i \le 4.6215886817230635 \cdot 10^{-236}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\left(i \cdot i\right) \cdot \left(50 + i \cdot \frac{50}{3}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.047641306012117 \cdot 10^{-225}:\\ \;\;\;\;\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.007511268653787123:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\left(i \cdot i\right) \cdot \left(50 + i \cdot \frac{50}{3}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.710926129998514 \cdot 10^{+244}:\\ \;\;\;\;\frac{\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot 50 - \left(\left(\frac{100}{3} \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log n \cdot \log i\right) \cdot \left(n \cdot n\right)\right) \cdot 100\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\log n \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \left(100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\log i \cdot \left(n \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(\log i \cdot n\right), 100, \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log i\right), \left(\mathsf{fma}\left(\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log i\right), \frac{50}{3}, \left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot 50\right)\right)\right)\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r4495132 = 100.0;
        double r4495133 = 1.0;
        double r4495134 = i;
        double r4495135 = n;
        double r4495136 = r4495134 / r4495135;
        double r4495137 = r4495133 + r4495136;
        double r4495138 = pow(r4495137, r4495135);
        double r4495139 = r4495138 - r4495133;
        double r4495140 = r4495139 / r4495136;
        double r4495141 = r4495132 * r4495140;
        return r4495141;
}

↓

double f(double i, double n) {
        double r4495142 = i;
        double r4495143 = -1.9035517225708853e-05;
        bool r4495144 = r4495142 <= r4495143;
        double r4495145 = 1.0;
        double r4495146 = r4495145 / r4495142;
        double r4495147 = n;
        double r4495148 = r4495142 / r4495147;
        double r4495149 = log1p(r4495148);
        double r4495150 = r4495149 * r4495147;
        double r4495151 = exp(r4495150);
        double r4495152 = 100.0;
        double r4495153 = -100.0;
        double r4495154 = fma(r4495151, r4495152, r4495153);
        double r4495155 = r4495154 * r4495147;
        double r4495156 = r4495146 * r4495155;
        double r4495157 = 4.6215886817230635e-236;
        bool r4495158 = r4495142 <= r4495157;
        double r4495159 = r4495142 * r4495142;
        double r4495160 = 50.0;
        double r4495161 = 16.666666666666668;
        double r4495162 = r4495142 * r4495161;
        double r4495163 = r4495160 + r4495162;
        double r4495164 = r4495159 * r4495163;
        double r4495165 = fma(r4495142, r4495152, r4495164);
        double r4495166 = r4495165 / r4495148;
        double r4495167 = 5.047641306012117e-225;
        bool r4495168 = r4495142 <= r4495167;
        double r4495169 = r4495145 + r4495148;
        double r4495170 = pow(r4495169, r4495147);
        double r4495171 = r4495152 * r4495170;
        double r4495172 = r4495171 + r4495153;
        double r4495173 = r4495172 / r4495148;
        double r4495174 = 0.007511268653787123;
        bool r4495175 = r4495142 <= r4495174;
        double r4495176 = 1.710926129998514e+244;
        bool r4495177 = r4495142 <= r4495176;
        double r4495178 = log(r4495147);
        double r4495179 = r4495147 * r4495178;
        double r4495180 = r4495179 * r4495179;
        double r4495181 = r4495180 * r4495160;
        double r4495182 = 33.333333333333336;
        double r4495183 = r4495147 * r4495147;
        double r4495184 = r4495179 * r4495183;
        double r4495185 = log(r4495142);
        double r4495186 = r4495185 * r4495185;
        double r4495187 = r4495184 * r4495186;
        double r4495188 = r4495182 * r4495187;
        double r4495189 = r4495178 * r4495185;
        double r4495190 = r4495189 * r4495183;
        double r4495191 = r4495190 * r4495152;
        double r4495192 = r4495188 + r4495191;
        double r4495193 = r4495180 * r4495147;
        double r4495194 = r4495178 * r4495193;
        double r4495195 = r4495152 * r4495179;
        double r4495196 = fma(r4495161, r4495187, r4495195);
        double r4495197 = fma(r4495161, r4495194, r4495196);
        double r4495198 = r4495192 + r4495197;
        double r4495199 = r4495181 - r4495198;
        double r4495200 = r4495185 * r4495147;
        double r4495201 = r4495200 * r4495200;
        double r4495202 = r4495147 * r4495201;
        double r4495203 = r4495185 * r4495202;
        double r4495204 = r4495193 * r4495185;
        double r4495205 = r4495201 * r4495160;
        double r4495206 = fma(r4495204, r4495161, r4495205);
        double r4495207 = fma(r4495182, r4495204, r4495206);
        double r4495208 = fma(r4495200, r4495152, r4495207);
        double r4495209 = fma(r4495161, r4495203, r4495208);
        double r4495210 = r4495199 + r4495209;
        double r4495211 = r4495210 / r4495148;
        double r4495212 = r4495177 ? r4495211 : r4495173;
        double r4495213 = r4495175 ? r4495166 : r4495212;
        double r4495214 = r4495168 ? r4495173 : r4495213;
        double r4495215 = r4495158 ? r4495166 : r4495214;
        double r4495216 = r4495144 ? r4495156 : r4495215;
        return r4495216;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	42.2
Herbie	25.2

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

Derivation

1. Split input into 4 regimes

2. if i < -1.9035517225708853e-05

   1. Initial program 29.3

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

   2. Simplified 29.2

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

   4. Applied pow-to-exp 29.2

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

   5. Simplified 5.8

      \[\leadsto \frac{\mathsf{fma}\left(100, \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}}\right), -100\right)}{\frac{i}{n}}\]
   6. Using strategy rm

   7. Applied div-inv 5.8

      \[\leadsto \frac{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}{\color{blue}{i \cdot \frac{1}{n}}}\]

   8. Applied *-un-lft-identity 5.8

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}}{i \cdot \frac{1}{n}}\]

   9. Applied times-frac 6.9

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \frac{\mathsf{fma}\left(100, \left(e^{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}\right), -100\right)}{\frac{1}{n}}}\]

   10. Simplified 6.8

       \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), 100, -100\right) \cdot n\right)}\]

   if -1.9035517225708853e-05 < i < 4.6215886817230635e-236 or 5.047641306012117e-225 < i < 0.007511268653787123

   1. Initial program 50.3

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

   2. Simplified 50.3

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

   3. Taylor expanded around 0 32.8

      \[\leadsto \frac{\color{blue}{100 \cdot i + \left(50 \cdot {i}^{2} + \frac{50}{3} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   4. Simplified 32.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(i, 100, \left(\left(i \cdot i\right) \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)\right)}}{\frac{i}{n}}\]

   if 4.6215886817230635e-236 < i < 5.047641306012117e-225 or 1.710926129998514e+244 < i

   1. Initial program 36.9

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

   2. Simplified 36.8

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

   4. Applied fma-udef 36.8

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

   if 0.007511268653787123 < i < 1.710926129998514e+244

   1. Initial program 33.8

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

   2. Simplified 33.8

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

   4. Applied pow-to-exp 48.0

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

   5. Simplified 46.8

      \[\leadsto \frac{\mathsf{fma}\left(100, \left(e^{\color{blue}{n \cdot \mathsf{log1p}\left(\left(\frac{i}{n}\right)\right)}}\right), -100\right)}{\frac{i}{n}}\]

   6. Taylor expanded around 0 16.3

      \[\leadsto \frac{\color{blue}{\left(50 \cdot \left({n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(100 \cdot \left(n \cdot \log i\right) + \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + 50 \cdot \left({n}^{2} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right)\right)\right) - \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log n \cdot \log i\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log i \cdot \log n\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + 100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]

   7. Simplified 16.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right) \cdot \log i\right), \left(\mathsf{fma}\left(\left(n \cdot \log i\right), 100, \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \log i\right), \left(\mathsf{fma}\left(\left(\left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \log i\right), \frac{50}{3}, \left(50 \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) + \left(50 \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) - \left(\left(\left(\left(\left(n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right) \cdot \frac{100}{3} + \left(\left(n \cdot n\right) \cdot \left(\log n \cdot \log i\right)\right) \cdot 100\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \log n\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \left(100 \cdot \left(\log n \cdot n\right)\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 25.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.9035517225708853 \cdot 10^{-05}:\\ \;\;\;\;\frac{1}{i} \cdot \left(\mathsf{fma}\left(\left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), 100, -100\right) \cdot n\right)\\ \mathbf{elif}\;i \le 4.6215886817230635 \cdot 10^{-236}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\left(i \cdot i\right) \cdot \left(50 + i \cdot \frac{50}{3}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 5.047641306012117 \cdot 10^{-225}:\\ \;\;\;\;\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.007511268653787123:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, 100, \left(\left(i \cdot i\right) \cdot \left(50 + i \cdot \frac{50}{3}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.710926129998514 \cdot 10^{+244}:\\ \;\;\;\;\frac{\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot 50 - \left(\left(\frac{100}{3} \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log n \cdot \log i\right) \cdot \left(n \cdot n\right)\right) \cdot 100\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\log n \cdot \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right)\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \log i\right)\right), \left(100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)\right) + \mathsf{fma}\left(\frac{50}{3}, \left(\log i \cdot \left(n \cdot \left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(\log i \cdot n\right), 100, \left(\mathsf{fma}\left(\frac{100}{3}, \left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log i\right), \left(\mathsf{fma}\left(\left(\left(\left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) \cdot n\right) \cdot \log i\right), \frac{50}{3}, \left(\left(\left(\log i \cdot n\right) \cdot \left(\log i \cdot n\right)\right) \cdot 50\right)\right)\right)\right)\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :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))))