Average Error: 47.2 → 0.3
Time: 35.7s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{i}{n} \le -118194.01059359145:\\ \;\;\;\;\left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;\frac{i}{n} \le -7.667481336184135 \cdot 10^{-303}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{i}{n} \le 2.4589229526522218 \cdot 10^{-297} \lor \neg \left(\frac{i}{n} \le 3.700705677166386 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot (e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{i}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.2
Target47.3
Herbie0.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ i n) < -118194.01059359145

    1. Initial program 0.4

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

    3. Applied associate-/r/0.4

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

    4. Applied associate-*r*0.4

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

    if -118194.01059359145 < (/ i n) < -7.667481336184135e-303

    1. Initial program 60.8

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

    3. Applied add-exp-log60.8

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

    4. Applied expm1-def60.8

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

    5. Simplified0.5

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

    if -7.667481336184135e-303 < (/ i n) < 2.4589229526522218e-297 or 3.700705677166386e+303 < (/ i n)

    1. Initial program 51.4

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

    3. Applied add-exp-log51.4

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

    4. Applied expm1-def51.4

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

    5. Simplified44.3

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

    7. Applied associate-*r/44.3

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

    9. Applied clear-num44.4

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

    10. Taylor expanded around 0 9.6

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

    11. Simplified0.3

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

    if 2.4589229526522218e-297 < (/ i n) < 3.700705677166386e+303

    1. Initial program 42.0

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

    3. Applied add-exp-log42.1

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

    4. Applied expm1-def42.1

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

    5. Simplified0.3

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

    7. Applied associate-*r/0.3

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

    9. Applied associate-/r/0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{i}{n} \le -118194.01059359145:\\ \;\;\;\;\left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;\frac{i}{n} \le -7.667481336184135 \cdot 10^{-303}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{i}{n} \le 2.4589229526522218 \cdot 10^{-297} \lor \neg \left(\frac{i}{n} \le 3.700705677166386 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot (e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{i}\\ \end{array}\]

Runtime

Time bar (total: 35.7s)Debug logProfile

herbie shell --seed 2018274 +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))))