Average Error: 42.6 → 15.4
Time: 36.0s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 0.0:\\ \;\;\;\;\frac{(e^{n \cdot (e^{\log_* (1 + \log_* (1 + \frac{i}{n}))} - 1)^*} - 1)^*}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 7.5484089451565 \cdot 10^{-07}:\\ \;\;\;\;\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(e^{n}\right)}^{\left(\log_* (1 + \frac{i}{n})\right)}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.6
Target42.7
Herbie15.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 0.0

    1. Initial program 39.9

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

    3. Applied add-exp-log39.9

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

    4. Applied pow-exp39.9

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

    5. Applied expm1-def31.9

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

    6. Simplified6.4

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

    8. Applied *-commutative6.4

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

    10. Applied expm1-log1p-u6.6

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

    if 0.0 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 7.5484089451565e-07

    1. Initial program 2.9

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

    2. Taylor expanded around inf 57.4

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

    3. Simplified2.9

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

    if 7.5484089451565e-07 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))

    1. Initial program 59.9

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

    3. Applied add-exp-log60.7

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

    4. Applied pow-exp60.7

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

    5. Applied expm1-def60.7

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

    6. Simplified60.7

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

    8. Applied *-commutative60.7

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

    10. Applied expm1-log1p-u60.7

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

    12. Applied expm1-udef60.7

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

    13. Applied div-sub60.7

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

    14. Simplified47.6

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 0.0:\\ \;\;\;\;\frac{(e^{n \cdot (e^{\log_* (1 + \log_* (1 + \frac{i}{n}))} - 1)^*} - 1)^*}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 7.5484089451565 \cdot 10^{-07}:\\ \;\;\;\;\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(e^{n}\right)}^{\left(\log_* (1 + \frac{i}{n})\right)}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \end{array}\]

Runtime

Time bar (total: 36.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes20.515.49.910.648.5%
herbie shell --seed 2018351 +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))))