Average Error: 42.7 → 29.3
Time: 6.6m
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.1392616853211877:\\ \;\;\;\;\left(\frac{n}{i} \cdot \left({\left(\frac{1}{\frac{n}{i}}\right)}^{n} - 1\right)\right) \cdot 100\\ \mathbf{elif}\;i \le 48.21334776389858:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.5
Herbie29.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.1392616853211877

    1. Initial program 27.9

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

    2. Taylor expanded around inf 62.9

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

    3. Simplified19.1

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

    if -2.1392616853211877 < i < 48.21334776389858

    1. Initial program 50.3

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

    2. Taylor expanded around 0 32.7

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

    3. Simplified32.7

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

    if 48.21334776389858 < i

    1. Initial program 30.8

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

    2. Taylor expanded around 0 30.2

      \[\leadsto \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification29.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.1392616853211877:\\ \;\;\;\;\left(\frac{n}{i} \cdot \left({\left(\frac{1}{\frac{n}{i}}\right)}^{n} - 1\right)\right) \cdot 100\\ \mathbf{elif}\;i \le 48.21334776389858:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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))))