Average Error: 47.2 → 5.6
Time: 27.3s
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -3.331964751214425 \cdot 10^{-84}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.1078131615865085 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{elif}\;i \le 2.6212041651656725 \cdot 10^{+101}:\\ \;\;\;\;n \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{i}\\ \mathbf{elif}\;i \le 1.4220196579526484 \cdot 10^{+233} \lor \neg \left(i \le 1.663230148188303 \cdot 10^{+280}\right):\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.2
Target47.3
Herbie5.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -3.331964751214425e-84

    1. Initial program 34.2

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

    3. Applied pow-to-exp34.3

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

    4. Applied expm1-def28.1

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

    5. Simplified0.9

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

    if -3.331964751214425e-84 < i < 1.1078131615865085e-66 or 1.4220196579526484e+233 < i < 1.663230148188303e+280

    1. Initial program 57.3

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

    3. Applied pow-to-exp58.7

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

    4. Applied expm1-def54.4

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

    5. Simplified20.8

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

    7. Applied associate-*r/20.9

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

    9. Applied clear-num21.0

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

    10. Taylor expanded around 0 11.0

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

    11. Simplified4.9

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

    if 1.1078131615865085e-66 < i < 2.6212041651656725e+101

    1. Initial program 45.6

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

    3. Applied pow-to-exp47.9

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

    4. Applied expm1-def33.0

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

    5. Simplified7.3

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

    7. Applied associate-*r/7.3

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

    9. Applied associate-/r/7.7

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

    if 2.6212041651656725e+101 < i < 1.4220196579526484e+233 or 1.663230148188303e+280 < i

    1. Initial program 27.8

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

    3. Applied associate-/r/27.8

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

    4. Applied associate-*r*27.7

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.331964751214425 \cdot 10^{-84}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.1078131615865085 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{elif}\;i \le 2.6212041651656725 \cdot 10^{+101}:\\ \;\;\;\;n \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{i}\\ \mathbf{elif}\;i \le 1.4220196579526484 \cdot 10^{+233} \lor \neg \left(i \le 1.663230148188303 \cdot 10^{+280}\right):\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 27.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes15.15.60.115.063.3%
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))))