Average Error: 47.5 → 11.8
Time: 2.8m
Precision: 64
Internal Precision: 3456
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(i \cdot \left(i \cdot \frac{1}{2}\right) + i)_*}{i} \le 0.9999999999999807:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;\frac{(i \cdot \left(i \cdot \frac{1}{2}\right) + i)_*}{i} \le 27145573243.50937:\\ \;\;\;\;\frac{(i \cdot \left(i \cdot \frac{1}{2}\right) + i)_*}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{if}\;\frac{(i \cdot \left(i \cdot \frac{1}{2}\right) + i)_*}{i} \le 5.20904361381548 \cdot 10^{+152}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{n}{i}\right) \cdot (e^{\frac{\log n - \log i}{n}} - 1)^*\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.5
Target47.1
Herbie11.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (fma i (* i 1/2) i) i) < 0.9999999999999807

    1. Initial program 29.2

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

    3. Applied add-exp-log29.3

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

    4. Applied pow-exp29.3

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

    5. Applied simplify6.4

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

    if 0.9999999999999807 < (/ (fma i (* i 1/2) i) i) < 27145573243.50937

    1. Initial program 57.6

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

    2. Taylor expanded around 0 57.4

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

    3. Applied simplify25.6

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

    5. Applied associate-/r/9.5

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

    if 27145573243.50937 < (/ (fma i (* i 1/2) i) i) < 5.20904361381548e+152

    1. Initial program 31.7

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

    3. Applied associate-/r/31.8

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

    4. Applied associate-*r*31.7

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

    if 5.20904361381548e+152 < (/ (fma i (* i 1/2) i) i)

    1. Initial program 33.2

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

    2. Taylor expanded around inf 32.8

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

    3. Applied simplify33.9

      \[\leadsto \color{blue}{\left(100 \cdot \frac{n}{i}\right) \cdot (e^{\frac{\log n - \log i}{n}} - 1)^*}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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))))