Average Error: 42.3 → 29.4
Time: 25.3s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -228.16046168847316:\\ \;\;\;\;\left({\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 0.959975051934705:\\ \;\;\;\;\frac{i \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + i}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) - 1}{\left(\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) + {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot \frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.3
Target42.4
Herbie29.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -228.16046168847316

    1. Initial program 26.6

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

    2. Taylor expanded around -inf 29.4

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

    3. Simplified18.4

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

    5. Applied associate-*r/18.4

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

    6. Taylor expanded around -inf 29.4

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

    7. Simplified18.4

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

    if -228.16046168847316 < i < 0.959975051934705

    1. Initial program 50.3

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

    2. Taylor expanded around 0 33.0

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

    3. Simplified33.0

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

    if 0.959975051934705 < i

    1. Initial program 30.8

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

    3. Applied flip3--30.9

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

    4. Applied associate-/l/30.9

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

    5. Simplified30.9

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

  4. Final simplification29.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -228.16046168847316:\\ \;\;\;\;\left({\left(\frac{1}{n} \cdot i\right)}^{n} \cdot 100 - 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;i \le 0.959975051934705:\\ \;\;\;\;\frac{i \cdot \left(i \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)\right) + i}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) - 1}{\left(\left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right) + {\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot \frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(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))))