Average Error: 42.6 → 18.9
Time: 33.7s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 5.7417296647533056 \cdot 10^{+103}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.6
Target42.3
Herbie18.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < 5.7417296647533056e+103

    1. Initial program 43.8

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

    3. Applied pow-to-exp44.1

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

    4. Applied expm1-def38.5

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

    5. Simplified17.5

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

    7. Applied associate-*r/17.6

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

    if 5.7417296647533056e+103 < i

    1. Initial program 31.5

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

    2. Taylor expanded around inf 29.4

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

    3. Simplified31.5

      \[\leadsto \color{blue}{\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 5.7417296647533056 \cdot 10^{+103}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \end{array}\]

Runtime

Time bar (total: 33.7s)Debug logProfile

herbie shell --seed 2018285 +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))))