Average Error: 47.7 → 14.5
Time: 13.2s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.016787435054206774:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.012023556076648468:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.016787435054206774:\\
\;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 0.012023556076648468:\\
\;\;\;\;100 \cdot \left(n \cdot \left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\

\end{array}
(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
↓
(FPCore (i n)
 :precision binary64
 (if (<= i -0.016787435054206774)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i 0.012023556076648468)
     (*
      100.0
      (*
       n
       (*
        (cbrt
         (/
          (+
           (* i 1.0)
           (+ (* n (log 1.0)) (* i (* i (- 0.5 (* (log 1.0) 0.5))))))
          i))
        (*
         (cbrt
          (/
           (+
            (* i 1.0)
            (+ (* n (log 1.0)) (* i (* i (- 0.5 (* (log 1.0) 0.5))))))
           i))
         (cbrt
          (/
           (+
            (* i 1.0)
            (+ (* n (log 1.0)) (* i (* i (- 0.5 (* (log 1.0) 0.5))))))
           i))))))
     (*
      100.0
      (/
       (/
        (- (pow (+ (/ i n) 1.0) (* n 2.0)) (* 1.0 1.0))
        (+ 1.0 (pow (+ (/ i n) 1.0) n)))
       (/ i n))))))
double code(double i, double n) {
	return ((double) (100.0 * (((double) (((double) pow(((double) (1.0 + (i / n))), n)) - 1.0)) / (i / n))));
}
↓
double code(double i, double n) {
	double tmp;
	if ((i <= -0.016787435054206774)) {
		tmp = ((double) (100.0 * (((double) (((double) pow((i / n), n)) - 1.0)) / (i / n))));
	} else {
		double tmp_1;
		if ((i <= 0.012023556076648468)) {
			tmp_1 = ((double) (100.0 * ((double) (n * ((double) (((double) cbrt((((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (i * ((double) (i * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))) / i))) * ((double) (((double) cbrt((((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (i * ((double) (i * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))) / i))) * ((double) cbrt((((double) (((double) (i * 1.0)) + ((double) (((double) (n * ((double) log(1.0)))) + ((double) (i * ((double) (i * ((double) (0.5 - ((double) (((double) log(1.0)) * 0.5)))))))))))) / i)))))))))));
		} else {
			tmp_1 = ((double) (100.0 * ((((double) (((double) pow(((double) ((i / n) + 1.0)), ((double) (n * 2.0)))) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + ((double) pow(((double) ((i / n) + 1.0)), n))))) / (i / n))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.7
Target47.0
Herbie14.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.016787435054206774

    1. Initial program Error: 28.6 bits

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

    2. Taylor expanded around inf Error: 64.0 bits

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

    3. SimplifiedError: 18.9 bits

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

    if -0.016787435054206774 < i < 0.0120235560766484683

    1. Initial program Error: 58.1 bits

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

    2. Taylor expanded around 0 Error: 26.8 bits

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{i}{n}}\]

    3. SimplifiedError: 26.8 bits

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied associate-/r/Error: 9.5 bits

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot \left(i \cdot 0.5\right) - \left(i \cdot i\right) \cdot \left(0.5 \cdot \log 1\right)\right)\right)}{i} \cdot n\right)}\]

    6. SimplifiedError: 9.5 bits

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}} \cdot n\right)\]
    7. Using strategy rm

    8. Applied add-cube-cbrtError: 9.5 bits

      \[\leadsto 100 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}} \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}\right) \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}\right)} \cdot n\right)\]

    9. SimplifiedError: 9.5 bits

      \[\leadsto 100 \cdot \left(\left(\color{blue}{\left(\sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}}\right)} \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + \left(i \cdot i\right) \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)}{i}}\right) \cdot n\right)\]

    10. SimplifiedError: 9.5 bits

      \[\leadsto 100 \cdot \left(\left(\left(\sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}}\right) \cdot \color{blue}{\sqrt[3]{\frac{1 \cdot i + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}}}\right) \cdot n\right)\]

    if 0.0120235560766484683 < i

    1. Initial program Error: 31.1 bits

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

    3. Applied flip--Error: 31.1 bits

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

    4. SimplifiedError: 31.1 bits

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

    5. SimplifiedError: 31.1 bits

      \[\leadsto 100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{\color{blue}{1 + {\left(1 + \frac{i}{n}\right)}^{n}}}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 14.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.016787435054206774:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.012023556076648468:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \left(\sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}} \cdot \sqrt[3]{\frac{i \cdot 1 + \left(n \cdot \log 1 + i \cdot \left(i \cdot \left(0.5 - \log 1 \cdot 0.5\right)\right)\right)}{i}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{\left(n \cdot 2\right)} - 1 \cdot 1}{1 + {\left(\frac{i}{n} + 1\right)}^{n}}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))