Average Error: 42.5 → 18.5
Time: 43.4s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -0.6742707270729336:\\ \;\;\;\;100 \cdot \frac{(e^{i} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8.035164529298766 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le -1.1799806447550313 \cdot 10^{-177}:\\ \;\;\;\;n \cdot \left(\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le 1.5420230324639676 \cdot 10^{-253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.5
Target42.3
Herbie18.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -0.6742707270729336

    1. Initial program 44.3

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

    3. Applied add-exp-log44.3

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

    4. Applied expm1-def44.3

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

    5. Simplified24.4

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

    7. Applied *-commutative24.4

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

    8. Taylor expanded around 0 24.4

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

    if -0.6742707270729336 < n < -8.035164529298766e-122 or -1.1799806447550313e-177 < n < 1.5420230324639676e-253

    1. Initial program 18.2

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

    3. Applied add-exp-log18.2

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

    4. Applied expm1-def18.2

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

    5. Simplified21.6

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

    6. Taylor expanded around 0 15.3

      \[\leadsto \color{blue}{0}\]

    if -8.035164529298766e-122 < n < -1.1799806447550313e-177

    1. Initial program 17.0

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

    3. Applied add-exp-log17.0

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

    4. Applied expm1-def17.0

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

    5. Simplified20.0

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

    7. Applied associate-/r/20.6

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

    8. Applied associate-*r*20.5

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

    if 1.5420230324639676e-253 < n

    1. Initial program 56.0

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

    3. Applied add-exp-log56.0

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

    4. Applied expm1-def56.0

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

    5. Simplified15.8

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

    7. Applied *-commutative15.8

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -0.6742707270729336:\\ \;\;\;\;100 \cdot \frac{(e^{i} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8.035164529298766 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le -1.1799806447550313 \cdot 10^{-177}:\\ \;\;\;\;n \cdot \left(\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le 1.5420230324639676 \cdot 10^{-253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100\\ \end{array}\]

Runtime

Time bar (total: 43.4s)Debug logProfile

herbie shell --seed 2018278 +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))))