Average Error: 47.0 → 5.8
Time: 37.1s
Precision: 64
Internal Precision: 3392
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.6644889723028783 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{\frac{i}{n}}}{\frac{\frac{1}{100}}{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}\\ \mathbf{elif}\;i \le 1.4017858767558676 \cdot 10^{-106} \lor \neg \left(i \le 8.656962291612699 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^* \cdot 100}{\frac{i}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original47.0
Target47.0
Herbie5.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.6644889723028783e-28

    1. Initial program 29.1

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

    3. Applied add-exp-log29.1

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

    4. Applied pow-exp29.1

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

    5. Applied expm1-def22.8

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

    6. Simplified0.4

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

    8. Applied associate-*r/0.3

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

    10. Applied clear-num0.6

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

    12. Applied div-inv0.6

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

    13. Applied associate-/r*0.5

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

    14. Simplified0.5

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

    if -1.6644889723028783e-28 < i < 1.4017858767558676e-106 or 8.656962291612699e+76 < i

    1. Initial program 53.9

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

    3. Applied add-exp-log57.8

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

    4. Applied pow-exp57.8

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

    5. Applied expm1-def52.9

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

    6. Simplified24.4

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

    8. Applied associate-*r/24.4

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

    10. Applied clear-num24.7

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

    11. Taylor expanded around 0 15.8

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

    12. Simplified7.8

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

    if 1.4017858767558676e-106 < i < 8.656962291612699e+76

    1. Initial program 49.2

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

    3. Applied add-exp-log50.6

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

    4. Applied pow-exp50.6

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

    5. Applied expm1-def38.7

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

    6. Simplified6.3

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

    8. Applied associate-*r/6.4

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.6644889723028783 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{\frac{i}{n}}}{\frac{\frac{1}{100}}{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}\\ \mathbf{elif}\;i \le 1.4017858767558676 \cdot 10^{-106} \lor \neg \left(i \le 8.656962291612699 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^* \cdot 100}{\frac{i}{n}}\\ \end{array}\]

Runtime

Time bar (total: 37.1s)Debug logProfile

herbie shell --seed 2018256 +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))))