Average Error: 42.3 → 15.0
Time: 53.4s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le -1.096990946805767 \cdot 10^{-241}:\\ \;\;\;\;\frac{\left(1 + \sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}} - 1\right) \cdot \left(\sqrt{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}} + 1\right)\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le -0.0:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\left(\frac{i}{n} - i\right) \cdot \frac{1}{2}}{n} + \frac{1}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 26.116748910123032:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\left(\frac{i}{n} - i\right) \cdot \frac{1}{2}}{n} + \frac{1}{n}}\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Target

Original42.3
Target42.1
Herbie15.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < -1.096990946805767e-241

    1. Initial program 2.2

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

    3. Applied add-sqr-sqrt2.3

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

    4. Applied difference-of-sqr-12.4

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}}{\frac{i}{n}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt2.4

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

    7. Applied sqrt-prod2.4

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

    8. Applied difference-of-sqr-12.4

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

    if -1.096990946805767e-241 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < -0.0 or 26.116748910123032 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))

    1. Initial program 46.8

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

    3. Applied add-sqr-sqrt46.8

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

    4. Applied difference-of-sqr-146.8

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}}{\frac{i}{n}}\]
    5. Using strategy rm

    6. Applied flip3--46.8

      \[\leadsto 100 \cdot \frac{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \color{blue}{\frac{{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^{3} - {1}^{3}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \left(1 \cdot 1 + \sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} \cdot 1\right)}}}{\frac{i}{n}}\]
    7. Using strategy rm

    8. Applied clear-num46.8

      \[\leadsto 100 \cdot \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \frac{{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^{3} - {1}^{3}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \left(1 \cdot 1 + \sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} \cdot 1\right)}}}}\]

    9. Taylor expanded around 0 20.3

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

    10. Simplified16.3

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

    if -0.0 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 26.116748910123032

    1. Initial program 3.0

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

    2. Taylor expanded around inf 57.6

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

    3. Simplified3.6

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le -1.096990946805767 \cdot 10^{-241}:\\ \;\;\;\;\frac{\left(1 + \sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}} - 1\right) \cdot \left(\sqrt{\sqrt{{\left(\frac{i}{n} + 1\right)}^{n}}} + 1\right)\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le -0.0:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\left(\frac{i}{n} - i\right) \cdot \frac{1}{2}}{n} + \frac{1}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 26.116748910123032:\\ \;\;\;\;\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\left(\frac{i}{n} - i\right) \cdot \frac{1}{2}}{n} + \frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(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))))