Average Error: 47.7 → 14.5
Time: 6.0m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{100}{i}}{i} \cdot \left(\frac{1}{2} + \frac{\frac{1}{6}}{i}\right) + i \cdot 100}{\frac{i}{n}} \le -1.2963324730749592 \cdot 10^{+173}:\\ \;\;\;\;100 \cdot \left(\left(\sqrt{\frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{i}} \cdot \left(\frac{5}{96} \cdot {i}^{2} + \left(1 + \frac{1}{4} \cdot i\right)\right)\right) \cdot n\right)\\ \mathbf{if}\;\frac{\frac{\frac{100}{i}}{i} \cdot \left(\frac{1}{2} + \frac{\frac{1}{6}}{i}\right) + i \cdot 100}{\frac{i}{n}} \le -5.923350227596516 \cdot 10^{-309}:\\ \;\;\;\;100 \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i} \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\right)\\ \mathbf{if}\;\frac{\frac{\frac{100}{i}}{i} \cdot \left(\frac{1}{2} + \frac{\frac{1}{6}}{i}\right) + i \cdot 100}{\frac{i}{n}} \le 0.0:\\ \;\;\;\;\frac{\frac{\frac{100}{i}}{i} \cdot \left(\frac{1}{2} + \frac{\frac{1}{6}}{i}\right) + i \cdot 100}{\frac{i}{n}}\\ \mathbf{if}\;\frac{\frac{\frac{100}{i}}{i} \cdot \left(\frac{1}{2} + \frac{\frac{1}{6}}{i}\right) + i \cdot 100}{\frac{i}{n}} \le 2.9110133691956703 \cdot 10^{+52}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - 1}{\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) + {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\sqrt{\frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{i}} \cdot \left(\frac{5}{96} \cdot {i}^{2} + \left(1 + \frac{1}{4} \cdot i\right)\right)\right) \cdot n\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.7
Target47.0
Herbie14.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (+ (* (/ (/ 100 i) i) (+ 1/2 (/ 1/6 i))) (* i 100)) (/ i n)) < -1.2963324730749592e+173 or 2.9110133691956703e+52 < (/ (+ (* (/ (/ 100 i) i) (+ 1/2 (/ 1/6 i))) (* i 100)) (/ i n))

    1. Initial program 57.8

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

    2. Taylor expanded around 0 28.5

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^{2} + \left(\frac{1}{6} \cdot {i}^{3} + i\right)}}{\frac{i}{n}}\]
    3. Using strategy rm

    4. Applied associate-/r/13.6

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

    5. Applied simplify13.6

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{i}} \cdot n\right)\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt13.6

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

    8. Taylor expanded around 0 13.6

      \[\leadsto 100 \cdot \left(\left(\sqrt{\frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{i}} \cdot \color{blue}{\left(\frac{5}{96} \cdot {i}^{2} + \left(1 + \frac{1}{4} \cdot i\right)\right)}\right) \cdot n\right)\]

    if -1.2963324730749592e+173 < (/ (+ (* (/ (/ 100 i) i) (+ 1/2 (/ 1/6 i))) (* i 100)) (/ i n)) < -5.923350227596516e-309

    1. Initial program 20.4

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

    3. Applied div-inv20.5

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

    4. Applied add-cube-cbrt20.5

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

    5. Applied times-frac20.5

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

    if -5.923350227596516e-309 < (/ (+ (* (/ (/ 100 i) i) (+ 1/2 (/ 1/6 i))) (* i 100)) (/ i n)) < 0.0

    1. Initial program 31.1

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

    2. Taylor expanded around 0 55.1

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

    3. Taylor expanded around inf 0.0

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

    4. Applied simplify0.0

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

    if 0.0 < (/ (+ (* (/ (/ 100 i) i) (+ 1/2 (/ 1/6 i))) (* i 100)) (/ i n)) < 2.9110133691956703e+52

    1. Initial program 28.3

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

    3. Applied flip3--28.3

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

    4. Applied simplify28.3

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

    5. Applied simplify28.3

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

Runtime

Time bar (total: 6.0m)Debug log

herbie shell --seed '#(991339738 1419949195 2842012120 4157638069 1320221275 2092628673)' 
(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))))