Average Error: 47.3 → 7.7
Time: 29.2s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -4.679491147320558 \cdot 10^{-179}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.094009342251299 \cdot 10^{-93}:\\ \;\;\;\;n \cdot \left(\frac{(e^{i} - 1)^*}{i} \cdot 100\right)\\ \mathbf{elif}\;i \le 3.220582053393252 \cdot 10^{+103}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot (\left(\frac{100}{i}\right) \cdot \left({\left(\frac{i}{n}\right)}^{n}\right) + \left(\frac{-100}{i}\right))_*\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.3
Target47.0
Herbie7.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -4.679491147320558e-179

    1. Initial program 40.1

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

    3. Applied add-exp-log40.1

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

    4. Applied expm1-def40.1

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

    5. Simplified4.1

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

    7. Applied associate-*r/4.1

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

    if -4.679491147320558e-179 < i < 6.094009342251299e-93

    1. Initial program 59.6

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

    3. Applied add-exp-log59.6

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

    4. Applied expm1-def59.6

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

    5. Simplified22.5

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

    7. Applied associate-/r/23.4

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

    8. Applied associate-*r*23.3

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

    9. Taylor expanded around 0 5.1

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

    if 6.094009342251299e-93 < i < 3.220582053393252e+103

    1. Initial program 46.7

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

    3. Applied add-exp-log46.7

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

    4. Applied expm1-def46.7

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

    5. Simplified8.6

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

    7. Applied associate-*r/8.6

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

    9. Applied associate-/l*8.9

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

    if 3.220582053393252e+103 < i

    1. Initial program 31.5

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

    3. Applied add-exp-log31.5

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

    4. Applied expm1-def31.5

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

    5. Simplified51.0

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

    7. Applied associate-/r/51.0

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

    8. Applied associate-*r*50.9

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

    9. Taylor expanded around inf 29.3

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

    10. Simplified31.5

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -4.679491147320558 \cdot 10^{-179}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.094009342251299 \cdot 10^{-93}:\\ \;\;\;\;n \cdot \left(\frac{(e^{i} - 1)^*}{i} \cdot 100\right)\\ \mathbf{elif}\;i \le 3.220582053393252 \cdot 10^{+103}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot (\left(\frac{100}{i}\right) \cdot \left({\left(\frac{i}{n}\right)}^{n}\right) + \left(\frac{-100}{i}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 29.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes16.17.70.016.052.1%
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))))