Compound Interest

Average Error: 43.1 → 21.0

Time: 20.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.070226988062184:\\ \;\;\;\;\frac{1}{\frac{i}{\left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot n}}\\ \mathbf{elif}\;i \le 1.1172319783923582 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(100 \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r145091 = 100.0;
        double r145092 = 1.0;
        double r145093 = i;
        double r145094 = n;
        double r145095 = r145093 / r145094;
        double r145096 = r145092 + r145095;
        double r145097 = pow(r145096, r145094);
        double r145098 = r145097 - r145092;
        double r145099 = r145098 / r145095;
        double r145100 = r145091 * r145099;
        return r145100;
}

↓

double f(double i, double n) {
        double r145101 = i;
        double r145102 = -1.070226988062184;
        bool r145103 = r145101 <= r145102;
        double r145104 = 1.0;
        double r145105 = 100.0;
        double r145106 = 1.0;
        double r145107 = n;
        double r145108 = r145101 / r145107;
        double r145109 = r145106 + r145108;
        double r145110 = pow(r145109, r145107);
        double r145111 = r145110 - r145106;
        double r145112 = r145105 * r145111;
        double r145113 = r145112 * r145107;
        double r145114 = r145101 / r145113;
        double r145115 = r145104 / r145114;
        double r145116 = 1.1172319783923582e-10;
        bool r145117 = r145101 <= r145116;
        double r145118 = r145106 * r145101;
        double r145119 = 0.5;
        double r145120 = 2.0;
        double r145121 = pow(r145101, r145120);
        double r145122 = r145119 * r145121;
        double r145123 = log(r145106);
        double r145124 = r145123 * r145107;
        double r145125 = r145122 + r145124;
        double r145126 = r145118 + r145125;
        double r145127 = r145121 * r145123;
        double r145128 = r145119 * r145127;
        double r145129 = r145126 - r145128;
        double r145130 = r145105 * r145129;
        double r145131 = r145130 * r145107;
        double r145132 = r145131 / r145101;
        double r145133 = r145110 * r145105;
        double r145134 = -r145106;
        double r145135 = r145134 * r145105;
        double r145136 = r145133 + r145135;
        double r145137 = r145136 / r145108;
        double r145138 = r145117 ? r145132 : r145137;
        double r145139 = r145103 ? r145115 : r145138;
        return r145139;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.4
Herbie	21.0

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

Derivation

1. Split input into 3 regimes

2. if i < -1.070226988062184

   1. Initial program 28.5

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

   3. Applied associate-*r/ 28.5

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

   5. Applied sub-neg 28.5

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

   6. Applied distribute-lft-in 28.5

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

   7. Simplified 28.5

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

   8. Simplified 28.5

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

   10. Applied *-un-lft-identity 28.5

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

   11. Applied *-un-lft-identity 28.5

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

   12. Applied times-frac 28.5

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

   13. Applied *-un-lft-identity 28.5

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

   14. Applied times-frac 28.5

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

   15. Simplified 28.5

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

   16. Simplified 29.0

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

   18. Applied clear-num 28.6

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

   if -1.070226988062184 < i < 1.1172319783923582e-10

   1. Initial program 51.0

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

   3. Applied associate-*r/ 51.0

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

   5. Applied sub-neg 51.0

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

   6. Applied distribute-lft-in 51.0

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

   7. Simplified 51.0

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

   8. Simplified 51.0

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

   10. Applied *-un-lft-identity 51.0

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

   11. Applied *-un-lft-identity 51.0

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

   12. Applied times-frac 51.0

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

   13. Applied *-un-lft-identity 51.0

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

   14. Applied times-frac 51.0

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

   15. Simplified 51.0

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

   16. Simplified 50.6

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

   17. Taylor expanded around 0 15.5

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

   if 1.1172319783923582e-10 < i

   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 associate-*r/ 32.8

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

   5. Applied sub-neg 32.8

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

   6. Applied distribute-lft-in 32.9

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

   7. Simplified 32.9

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

   8. Simplified 32.9

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

4. Final simplification 21.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.070226988062184:\\ \;\;\;\;\frac{1}{\frac{i}{\left(100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot n}}\\ \mathbf{elif}\;i \le 1.1172319783923582 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(100 \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))))