Average Error: 46.9 → 3.2
Time: 3.1m
Precision: 64
Internal Precision: 3136
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot n\right) \cdot \left(\frac{\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot \sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}{i} \cdot 100\right) \le -2.4209216646221 \cdot 10^{-322}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\left(\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot n\right) \cdot \left(\frac{\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot \sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}{i} \cdot 100\right) \le 2.926476725111146 \cdot 10^{-306}:\\ \;\;\;\;100 \cdot \frac{1}{(\frac{1}{2} \cdot \left(\frac{\frac{i}{n}}{n}\right) + \left(\frac{1}{n}\right))_* - \frac{1}{2} \cdot \frac{i}{n}}\\ \mathbf{elif}\;\left(\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot n\right) \cdot \left(\frac{\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot \sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}{i} \cdot 100\right) \le 1.776463078747528 \cdot 10^{+308}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot 0\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original46.9
Target46.5
Herbie3.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* 100 (/ (* (cbrt (expm1 (* (log1p (/ i n)) n))) (cbrt (expm1 (* (log1p (/ i n)) n)))) i)) (* (cbrt (expm1 (* (log1p (/ i n)) n))) n)) < -2.4209216646221e-322 or 2.926476725111146e-306 < (* (* 100 (/ (* (cbrt (expm1 (* (log1p (/ i n)) n))) (cbrt (expm1 (* (log1p (/ i n)) n)))) i)) (* (cbrt (expm1 (* (log1p (/ i n)) n))) n)) < 1.776463078747528e+308

    1. Initial program 54.6

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

    2. Initial simplification0.2

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

    if -2.4209216646221e-322 < (* (* 100 (/ (* (cbrt (expm1 (* (log1p (/ i n)) n))) (cbrt (expm1 (* (log1p (/ i n)) n)))) i)) (* (cbrt (expm1 (* (log1p (/ i n)) n))) n)) < 2.926476725111146e-306

    1. Initial program 35.8

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

    2. Initial simplification32.2

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

    4. Applied clear-num32.7

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

    5. Taylor expanded around 0 3.8

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

    6. Simplified1.8

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

    if 1.776463078747528e+308 < (* (* 100 (/ (* (cbrt (expm1 (* (log1p (/ i n)) n))) (cbrt (expm1 (* (log1p (/ i n)) n)))) i)) (* (cbrt (expm1 (* (log1p (/ i n)) n))) n))

    1. Initial program 32.9

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

    2. Initial simplification62.7

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

    3. Taylor expanded around 0 29.7

      \[\leadsto 100 \cdot \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot n\right) \cdot \left(\frac{\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot \sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}{i} \cdot 100\right) \le -2.4209216646221 \cdot 10^{-322}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\left(\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot n\right) \cdot \left(\frac{\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot \sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}{i} \cdot 100\right) \le 2.926476725111146 \cdot 10^{-306}:\\ \;\;\;\;100 \cdot \frac{1}{(\frac{1}{2} \cdot \left(\frac{\frac{i}{n}}{n}\right) + \left(\frac{1}{n}\right))_* - \frac{1}{2} \cdot \frac{i}{n}}\\ \mathbf{elif}\;\left(\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot n\right) \cdot \left(\frac{\sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*} \cdot \sqrt[3]{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}{i} \cdot 100\right) \le 1.776463078747528 \cdot 10^{+308}:\\ \;\;\;\;\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot 0\\ \end{array}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018214 +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))))