Compound Interest

Average Error: 42.2 → 12.5

Time: 28.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.3007934666970692 \cdot 10^{+19}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.525407785233792 \cdot 10^{-06}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{1}{100}}{n} - \frac{i}{n} \cdot \frac{1}{200}\right) + \frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{i}{n} \cdot \frac{1}{200}\right)\right)\right)\right)}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(50, \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right), \left(\mathsf{fma}\left(\left(\log i \cdot \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right)\right), \frac{50}{3}, \left(\mathsf{fma}\left(\left(n \cdot \log i\right), 100, \left(50 \cdot \left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log i\right) + 50 \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right) - \left(\left(\left(\log n \cdot \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right)\right) \cdot \frac{100}{3} + \mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log n\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log n \cdot \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right)\right), \left(100 \cdot \left(\log n \cdot n\right)\right)\right)\right)\right)\right) + 100 \cdot \left(\left(\log i \cdot \log n\right) \cdot \left(n \cdot n\right)\right)\right)}}\\ \end{array}\]

C

double f(double i, double n) {
        double r4110102 = 100.0;
        double r4110103 = 1.0;
        double r4110104 = i;
        double r4110105 = n;
        double r4110106 = r4110104 / r4110105;
        double r4110107 = r4110103 + r4110106;
        double r4110108 = pow(r4110107, r4110105);
        double r4110109 = r4110108 - r4110103;
        double r4110110 = r4110109 / r4110106;
        double r4110111 = r4110102 * r4110110;
        return r4110111;
}

↓

double f(double i, double n) {
        double r4110112 = i;
        double r4110113 = -1.3007934666970692e+19;
        bool r4110114 = r4110112 <= r4110113;
        double r4110115 = 100.0;
        double r4110116 = n;
        double r4110117 = r4110112 / r4110116;
        double r4110118 = log1p(r4110117);
        double r4110119 = r4110118 * r4110116;
        double r4110120 = exp(r4110119);
        double r4110121 = -100.0;
        double r4110122 = fma(r4110115, r4110120, r4110121);
        double r4110123 = exp(r4110122);
        double r4110124 = log(r4110123);
        double r4110125 = r4110124 / r4110117;
        double r4110126 = 9.525407785233792e-06;
        bool r4110127 = r4110112 <= r4110126;
        double r4110128 = 1.0;
        double r4110129 = 0.01;
        double r4110130 = r4110129 / r4110116;
        double r4110131 = 0.005;
        double r4110132 = r4110117 * r4110131;
        double r4110133 = r4110130 - r4110132;
        double r4110134 = log1p(r4110132);
        double r4110135 = expm1(r4110134);
        double r4110136 = r4110135 / r4110116;
        double r4110137 = r4110133 + r4110136;
        double r4110138 = r4110128 / r4110137;
        double r4110139 = 50.0;
        double r4110140 = log(r4110116);
        double r4110141 = r4110140 * r4110116;
        double r4110142 = r4110141 * r4110141;
        double r4110143 = log(r4110112);
        double r4110144 = r4110116 * r4110143;
        double r4110145 = r4110144 * r4110144;
        double r4110146 = r4110145 * r4110116;
        double r4110147 = r4110143 * r4110146;
        double r4110148 = 16.666666666666668;
        double r4110149 = r4110142 * r4110116;
        double r4110150 = r4110149 * r4110143;
        double r4110151 = r4110139 * r4110150;
        double r4110152 = r4110139 * r4110145;
        double r4110153 = r4110151 + r4110152;
        double r4110154 = fma(r4110144, r4110115, r4110153);
        double r4110155 = fma(r4110147, r4110148, r4110154);
        double r4110156 = fma(r4110139, r4110142, r4110155);
        double r4110157 = r4110140 * r4110146;
        double r4110158 = 33.333333333333336;
        double r4110159 = r4110157 * r4110158;
        double r4110160 = r4110149 * r4110140;
        double r4110161 = r4110115 * r4110141;
        double r4110162 = fma(r4110148, r4110157, r4110161);
        double r4110163 = fma(r4110148, r4110160, r4110162);
        double r4110164 = r4110159 + r4110163;
        double r4110165 = r4110143 * r4110140;
        double r4110166 = r4110116 * r4110116;
        double r4110167 = r4110165 * r4110166;
        double r4110168 = r4110115 * r4110167;
        double r4110169 = r4110164 + r4110168;
        double r4110170 = r4110156 - r4110169;
        double r4110171 = r4110117 / r4110170;
        double r4110172 = r4110128 / r4110171;
        double r4110173 = r4110127 ? r4110138 : r4110172;
        double r4110174 = r4110114 ? r4110125 : r4110173;
        return r4110174;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.2
Target	41.8
Herbie	12.5

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

Derivation

1. Split input into 3 regimes

2. if i < -1.3007934666970692e+19

   1. Initial program 26.5

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

   2. Simplified 26.5

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

   4. Applied add-exp-log 26.5

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

   5. Applied pow-exp 26.5

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

   6. Simplified 4.9

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

   8. Applied add-log-exp 4.9

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

   if -1.3007934666970692e+19 < i < 9.525407785233792e-06

   1. Initial program 49.6

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

   2. Simplified 49.6

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

   4. Applied add-exp-log 49.6

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

   5. Applied pow-exp 49.6

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

   6. Simplified 47.7

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

   8. Applied *-un-lft-identity 47.7

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

   9. Applied associate-/l* 47.7

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

   10. Taylor expanded around 0 14.5

       \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{200} \cdot \frac{i}{{n}^{2}} + \frac{1}{100} \cdot \frac{1}{n}\right) - \frac{1}{200} \cdot \frac{i}{n}}}\]

   11. Simplified 13.1

       \[\leadsto \frac{1}{\color{blue}{\left(\frac{\frac{1}{100}}{n} - \frac{1}{200} \cdot \frac{i}{n}\right) + \frac{\frac{1}{200} \cdot \frac{i}{n}}{n}}}\]
   12. Using strategy rm

   13. Applied expm1-log1p-u 13.1

       \[\leadsto \frac{1}{\left(\frac{\frac{1}{100}}{n} - \frac{1}{200} \cdot \frac{i}{n}\right) + \frac{\color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{1}{200} \cdot \frac{i}{n}\right)\right)\right)\right)}}{n}}\]

   if 9.525407785233792e-06 < i

   1. Initial program 32.0

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

   2. Simplified 31.9

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

   4. Applied add-exp-log 50.0

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

   5. Applied pow-exp 50.0

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

   6. Simplified 48.6

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

   8. Applied *-un-lft-identity 48.6

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

   9. Applied associate-/l* 48.6

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

   10. Taylor expanded around 0 20.7

       \[\leadsto \frac{1}{\frac{\frac{i}{n}}{\color{blue}{\left(50 \cdot \left({n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(100 \cdot \left(n \cdot \log i\right) + \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + 50 \cdot \left({n}^{2} \cdot {\left(\log i\right)}^{2}\right)\right)\right)\right)\right)\right) - \left(\frac{100}{3} \cdot \left({n}^{3} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log n \cdot \log i\right)\right) + \left(50 \cdot \left({n}^{2} \cdot \left(\log i \cdot \log n\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{50}{3} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + 100 \cdot \left(n \cdot \log n\right)\right)\right)\right)\right)\right)}}}\]

   11. Simplified 20.7

       \[\leadsto \frac{1}{\frac{\frac{i}{n}}{\color{blue}{\mathsf{fma}\left(50, \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right), \left(\mathsf{fma}\left(\left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right), \frac{50}{3}, \left(\mathsf{fma}\left(\left(n \cdot \log i\right), 100, \left(50 \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) + \left(\left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \log i\right) \cdot 50\right)\right)\right)\right)\right)\right) - \left(\left(\left(n \cdot n\right) \cdot \left(\log i \cdot \log n\right)\right) \cdot 100 + \left(\mathsf{fma}\left(\frac{50}{3}, \left(\left(n \cdot \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right)\right) \cdot \log n\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log n \cdot \left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right), \left(\left(\log n \cdot n\right) \cdot 100\right)\right)\right)\right) + \left(\log n \cdot \left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right) \cdot \frac{100}{3}\right)\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.3007934666970692 \cdot 10^{+19}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(100, \left(e^{\mathsf{log1p}\left(\left(\frac{i}{n}\right)\right) \cdot n}\right), -100\right)}\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 9.525407785233792 \cdot 10^{-06}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{1}{100}}{n} - \frac{i}{n} \cdot \frac{1}{200}\right) + \frac{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{i}{n} \cdot \frac{1}{200}\right)\right)\right)\right)}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(50, \left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right), \left(\mathsf{fma}\left(\left(\log i \cdot \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right)\right), \frac{50}{3}, \left(\mathsf{fma}\left(\left(n \cdot \log i\right), 100, \left(50 \cdot \left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log i\right) + 50 \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right)\right)\right)\right)\right) - \left(\left(\left(\log n \cdot \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right)\right) \cdot \frac{100}{3} + \mathsf{fma}\left(\frac{50}{3}, \left(\left(\left(\left(\log n \cdot n\right) \cdot \left(\log n \cdot n\right)\right) \cdot n\right) \cdot \log n\right), \left(\mathsf{fma}\left(\frac{50}{3}, \left(\log n \cdot \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right)\right), \left(100 \cdot \left(\log n \cdot n\right)\right)\right)\right)\right)\right) + 100 \cdot \left(\left(\log i \cdot \log n\right) \cdot \left(n \cdot n\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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))))