Average Error: 42.7 → 11.0
Time: 21.7s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.8110044447062212 \cdot 10^{-54}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 3.366692169582882 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{elif}\;i \le 5.000437644817621 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot (\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{n}{i}\right) + \left(-\frac{n}{i}\right))_*\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target41.6
Herbie11.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.8110044447062212e-54

    1. Initial program 33.3

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

    3. Applied add-exp-log33.3

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

    4. Applied expm1-def33.3

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

    5. Simplified0.6

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

    7. Applied associate-*r/0.5

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

    if -1.8110044447062212e-54 < i < 3.366692169582882e-66

    1. Initial program 49.0

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

    3. Applied add-exp-log49.0

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

    4. Applied expm1-def49.0

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

    5. Simplified28.2

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

    7. Applied associate-*r/28.3

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

    9. Applied clear-num28.5

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

    10. Taylor expanded around 0 14.2

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{200} \cdot \frac{i}{{n}^{2}} + \frac{1}{100} \cdot \frac{1}{n}\right) - \frac{1}{200} \cdot \frac{i}{n}}}\]

    11. Simplified12.8

      \[\leadsto \frac{1}{\color{blue}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}}\]

    if 3.366692169582882e-66 < i < 5.000437644817621e+132

    1. Initial program 44.2

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

    3. Applied add-exp-log44.2

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

    4. Applied expm1-def44.2

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

    5. Simplified10.8

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

    7. Applied associate-*r/10.8

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

    9. Applied clear-num11.1

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

    if 5.000437644817621e+132 < i

    1. Initial program 34.1

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

    3. Applied add-exp-log34.1

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

    4. Applied expm1-def34.1

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

    5. Simplified53.1

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

    6. Taylor expanded around inf 28.2

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

    7. Simplified34.0

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.8110044447062212 \cdot 10^{-54}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 3.366692169582882 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{elif}\;i \le 5.000437644817621 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot (\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot \left(\frac{n}{i}\right) + \left(-\frac{n}{i}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +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))))