Compound Interest

Average Error: 42.9 → 22.0

Time: 46.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.496898897117080818951261662080240322195 \cdot 10^{97}:\\ \;\;\;\;\left(n \cdot \left(\left(\sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -3.085998543664637255444788181328484588588 \cdot 10^{59}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \le -2.461738487841221623142473617917858064175:\\ \;\;\;\;\left(n \cdot \left(\left(\sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -3.388139377181462070651587205567840486657 \cdot 10^{-307}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 4.930404890743995467239903225253781019029 \cdot 10^{-103}:\\ \;\;\;\;100 \cdot \frac{\left(\left(1 + i \cdot 1\right) + \log 1 \cdot n\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \left(\left(\sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right)\right) \cdot 100\\ \end{array}\]

C

double f(double i, double n) {
        double r7466940 = 100.0;
        double r7466941 = 1.0;
        double r7466942 = i;
        double r7466943 = n;
        double r7466944 = r7466942 / r7466943;
        double r7466945 = r7466941 + r7466944;
        double r7466946 = pow(r7466945, r7466943);
        double r7466947 = r7466946 - r7466941;
        double r7466948 = r7466947 / r7466944;
        double r7466949 = r7466940 * r7466948;
        return r7466949;
}

↓

double f(double i, double n) {
        double r7466950 = n;
        double r7466951 = -1.4968988971170808e+97;
        bool r7466952 = r7466950 <= r7466951;
        double r7466953 = 1.0;
        double r7466954 = log(r7466953);
        double r7466955 = r7466954 * r7466950;
        double r7466956 = i;
        double r7466957 = r7466956 * r7466953;
        double r7466958 = r7466956 * r7466956;
        double r7466959 = 0.5;
        double r7466960 = r7466958 * r7466959;
        double r7466961 = r7466957 + r7466960;
        double r7466962 = r7466954 * r7466960;
        double r7466963 = r7466961 - r7466962;
        double r7466964 = r7466955 + r7466963;
        double r7466965 = r7466964 / r7466956;
        double r7466966 = cbrt(r7466965);
        double r7466967 = r7466966 * r7466966;
        double r7466968 = r7466967 * r7466966;
        double r7466969 = r7466950 * r7466968;
        double r7466970 = 100.0;
        double r7466971 = r7466969 * r7466970;
        double r7466972 = -3.0859985436646373e+59;
        bool r7466973 = r7466950 <= r7466972;
        double r7466974 = r7466956 / r7466950;
        double r7466975 = r7466974 + r7466953;
        double r7466976 = pow(r7466975, r7466950);
        double r7466977 = r7466976 - r7466953;
        double r7466978 = r7466977 / r7466956;
        double r7466979 = r7466978 * r7466970;
        double r7466980 = r7466979 * r7466950;
        double r7466981 = -2.4617384878412216;
        bool r7466982 = r7466950 <= r7466981;
        double r7466983 = -3.388139377181462e-307;
        bool r7466984 = r7466950 <= r7466983;
        double r7466985 = r7466976 / r7466974;
        double r7466986 = r7466953 / r7466974;
        double r7466987 = r7466985 - r7466986;
        double r7466988 = r7466970 * r7466987;
        double r7466989 = 4.930404890743995e-103;
        bool r7466990 = r7466950 <= r7466989;
        double r7466991 = 1.0;
        double r7466992 = r7466991 + r7466957;
        double r7466993 = r7466992 + r7466955;
        double r7466994 = r7466993 - r7466953;
        double r7466995 = r7466994 / r7466974;
        double r7466996 = r7466970 * r7466995;
        double r7466997 = r7466990 ? r7466996 : r7466971;
        double r7466998 = r7466984 ? r7466988 : r7466997;
        double r7466999 = r7466982 ? r7466971 : r7466998;
        double r7467000 = r7466973 ? r7466980 : r7466999;
        double r7467001 = r7466952 ? r7466971 : r7467000;
        return r7467001;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	43.1
Herbie	22.0

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

Derivation

1. Split input into 4 regimes

2. if n < -1.4968988971170808e+97 or -3.0859985436646373e+59 < n < -2.4617384878412216 or 4.930404890743995e-103 < n

   1. Initial program 53.8

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

   2. Taylor expanded around 0 39.5

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

   3. Simplified 39.5

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

   5. Applied associate-/r/ 20.8

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

   7. Applied add-cube-cbrt 20.9

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

   if -1.4968988971170808e+97 < n < -3.0859985436646373e+59

   1. Initial program 36.2

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

   3. Applied associate-/r/ 36.1

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

   4. Applied associate-*r* 36.1

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

   if -2.4617384878412216 < n < -3.388139377181462e-307

   1. Initial program 14.9

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

   3. Applied div-sub 15.0

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

   if -3.388139377181462e-307 < n < 4.930404890743995e-103

   1. Initial program 44.9

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

   2. Taylor expanded around 0 34.1

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

4. Final simplification 22.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.496898897117080818951261662080240322195 \cdot 10^{97}:\\ \;\;\;\;\left(n \cdot \left(\left(\sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -3.085998543664637255444788181328484588588 \cdot 10^{59}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \le -2.461738487841221623142473617917858064175:\\ \;\;\;\;\left(n \cdot \left(\left(\sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -3.388139377181462070651587205567840486657 \cdot 10^{-307}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 4.930404890743995467239903225253781019029 \cdot 10^{-103}:\\ \;\;\;\;100 \cdot \frac{\left(\left(1 + i \cdot 1\right) + \log 1 \cdot n\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \left(\left(\sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{\log 1 \cdot n + \left(\left(i \cdot 1 + \left(i \cdot i\right) \cdot 0.5\right) - \log 1 \cdot \left(\left(i \cdot i\right) \cdot 0.5\right)\right)}{i}}\right)\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))