Average Error: 47.3 → 14.3
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\log \left(e^{\sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}\right) \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}{i} \cdot \left(n \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}\right)\right) \cdot 100 \le -7.876677038040007 \cdot 10^{+303}:\\ \;\;\;\;\frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\left(\frac{\log \left(e^{\sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}\right) \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}{i} \cdot \left(n \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}\right)\right) \cdot 100 \le 1.7737416257178375 \cdot 10^{+308}:\\ \;\;\;\;100 \cdot n + \left(50 + \frac{50}{3} \cdot i\right) \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} - 1\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.3
Target46.8
Herbie14.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (* 100 (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (log (exp (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2))))))) i) (* n (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i))))))) < -7.876677038040007e+303

    1. Initial program 16.8

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

    3. Applied add-log-exp16.8

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

    if -7.876677038040007e+303 < (* 100 (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (log (exp (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2))))))) i) (* n (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i))))))) < 1.7737416257178375e+308

    1. Initial program 57.5

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

    2. Taylor expanded around 0 25.7

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

    3. Simplified25.7

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

    4. Taylor expanded around inf 9.8

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

    5. Simplified9.8

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

    if 1.7737416257178375e+308 < (* 100 (* (/ (* (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2)))) (log (exp (cbrt (+ i (* (* i i) (+ (* 1/6 i) 1/2))))))) i) (* n (cbrt (+ i (* (+ (* i 1/6) 1/2) (* i i)))))))

    1. Initial program 48.1

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

    2. Taylor expanded around inf 29.1

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

    3. Simplified31.4

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

  4. Final simplification14.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\log \left(e^{\sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}\right) \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}{i} \cdot \left(n \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}\right)\right) \cdot 100 \le -7.876677038040007 \cdot 10^{+303}:\\ \;\;\;\;\frac{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\left(\frac{\log \left(e^{\sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}\right) \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}}{i} \cdot \left(n \cdot \sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i}\right)\right) \cdot 100 \le 1.7737416257178375 \cdot 10^{+308}:\\ \;\;\;\;100 \cdot n + \left(50 + \frac{50}{3} \cdot i\right) \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} - 1\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018225 
(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))))