Average Error: 47.7 → 9.7
Time: 1.9m
Precision: 64
Internal Precision: 3200
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.012581553855926686:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.00033336288163752907:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{(i \cdot \left(\frac{1}{2} \cdot i\right) + i)_*}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{n}{i}\right) \cdot (e^{\left(\log i - \log n\right) \cdot n} - 1)^*\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.7
Target47.1
Herbie9.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.012581553855926686

    1. Initial program 28.2

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

    3. Applied add-exp-log28.2

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

    4. Applied pow-exp28.2

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

    5. Applied simplify5.4

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

    if -0.012581553855926686 < i < 0.00033336288163752907

    1. Initial program 57.9

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

    2. Taylor expanded around 0 57.5

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

    3. Applied simplify25.4

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

    5. Applied associate-/r/9.1

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

    if 0.00033336288163752907 < i

    1. Initial program 33.0

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

    2. Taylor expanded around inf 28.4

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

    3. Applied simplify19.4

      \[\leadsto \color{blue}{\left(100 \cdot \frac{n}{i}\right) \cdot (e^{n \cdot \left(0 + \left(\log i - \log n\right)\right)} - 1)^*}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify9.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -0.012581553855926686:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.00033336288163752907:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{(i \cdot \left(\frac{1}{2} \cdot i\right) + i)_*}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{n}{i}\right) \cdot (e^{\left(\log i - \log n\right) \cdot n} - 1)^*\\ \end{array}}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(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))))