Average Error: 47.6 → 10.4
Time: 15.6s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -21315269.074093234:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.02161134565546455:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;i \leq 4.3408139321053756 \cdot 10^{+272}:\\ \;\;\;\;\begin{array}{l} t_0 := {\log n}^{2}\\ t_1 := {\log i}^{2}\\ 100 \cdot \left(\left(\frac{\log i \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot t_1}{i} + \left(\frac{\log i \cdot {n}^{2}}{i} + \left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4} \cdot \left(\log i \cdot t_0\right)}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + 0.5 \cdot \frac{{n}^{3} \cdot t_0}{i}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{4} \cdot \left(t_1 \cdot \log n\right)}{i} + \left(\frac{{n}^{4} \cdot \log n}{{i}^{2}} + \left(\frac{{n}^{2} \cdot \log n}{i} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + 0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log n}^{3}}{i}\right)\right)\right)\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(\frac{i}{n} + 1\right)}^{n}, -100\right)}{i}\\ \end{array} \]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -21315269.074093234:\\
\;\;\;\;100 \cdot \frac{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 0.02161134565546455:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\

\mathbf{elif}\;i \leq 4.3408139321053756 \cdot 10^{+272}:\\
\;\;\;\;\begin{array}{l}
t_0 := {\log n}^{2}\\
t_1 := {\log i}^{2}\\
100 \cdot \left(\left(\frac{\log i \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot t_1}{i} + \left(\frac{\log i \cdot {n}^{2}}{i} + \left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4} \cdot \left(\log i \cdot t_0\right)}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + 0.5 \cdot \frac{{n}^{3} \cdot t_0}{i}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{4} \cdot \left(t_1 \cdot \log n\right)}{i} + \left(\frac{{n}^{4} \cdot \log n}{{i}^{2}} + \left(\frac{{n}^{2} \cdot \log n}{i} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + 0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log n}^{3}}{i}\right)\right)\right)\right)\right)\right)
\end{array}\\

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


\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 -21315269.074093234)
   (* 100.0 (/ (- (exp (* n (log1p (/ i n)))) 1.0) (/ i n)))
   (if (<= i 0.02161134565546455)
     (* 100.0 (* n (/ (expm1 i) i)))
     (if (<= i 4.3408139321053756e+272)
       (let* ((t_0 (pow (log n) 2.0)) (t_1 (pow (log i) 2.0)))
         (*
          100.0
          (-
           (+
            (/ (* (log i) (pow n 4.0)) (pow i 2.0))
            (+
             (* 0.5 (/ (* (pow n 3.0) t_1) i))
             (+
              (/ (* (log i) (pow n 2.0)) i)
              (+
               (/ (pow n 3.0) (pow i 2.0))
               (+
                (* 0.5 (/ (* (pow n 4.0) (* (log i) t_0)) i))
                (+
                 (*
                  0.16666666666666666
                  (/ (* (pow n 4.0) (pow (log i) 3.0)) i))
                 (* 0.5 (/ (* (pow n 3.0) t_0) i))))))))
           (+
            (* 0.5 (/ (* (pow n 4.0) (* t_1 (log n))) i))
            (+
             (/ (* (pow n 4.0) (log n)) (pow i 2.0))
             (+
              (/ (* (pow n 2.0) (log n)) i)
              (+
               (* 0.5 (/ (pow n 4.0) (pow i 3.0)))
               (+
                (/ (* (log i) (* (pow n 3.0) (log n))) i)
                (*
                 0.16666666666666666
                 (/ (* (pow n 4.0) (pow (log n) 3.0)) i))))))))))
       (* n (/ (fma 100.0 (pow (+ (/ i n) 1.0) n) -100.0) i))))))
double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}
↓
double code(double i, double n) {
	double tmp;
	if (i <= -21315269.074093234) {
		tmp = 100.0 * ((exp(n * log1p(i / n)) - 1.0) / (i / n));
	} else if (i <= 0.02161134565546455) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (i <= 4.3408139321053756e+272) {
		double t_0 = pow(log(n), 2.0);
		double t_1 = pow(log(i), 2.0);
		tmp = 100.0 * ((((log(i) * pow(n, 4.0)) / pow(i, 2.0)) + ((0.5 * ((pow(n, 3.0) * t_1) / i)) + (((log(i) * pow(n, 2.0)) / i) + ((pow(n, 3.0) / pow(i, 2.0)) + ((0.5 * ((pow(n, 4.0) * (log(i) * t_0)) / i)) + ((0.16666666666666666 * ((pow(n, 4.0) * pow(log(i), 3.0)) / i)) + (0.5 * ((pow(n, 3.0) * t_0) / i)))))))) - ((0.5 * ((pow(n, 4.0) * (t_1 * log(n))) / i)) + (((pow(n, 4.0) * log(n)) / pow(i, 2.0)) + (((pow(n, 2.0) * log(n)) / i) + ((0.5 * (pow(n, 4.0) / pow(i, 3.0))) + (((log(i) * (pow(n, 3.0) * log(n))) / i) + (0.16666666666666666 * ((pow(n, 4.0) * pow(log(n), 3.0)) / i))))))));
	} else {
		tmp = n * (fma(100.0, pow(((i / n) + 1.0), n), -100.0) / i);
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original47.6
Target47.6
Herbie10.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \]

Derivation

  1. Split input into 4 regimes

  2. if i < -21315269.0740932338

    1. Initial program 27.1

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

    2. Applied pow-to-exp_binary6427.1

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

    3. Simplified5.9

      \[\leadsto 100 \cdot \frac{e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}} - 1}{\frac{i}{n}} \]

    if -21315269.0740932338 < i < 0.0216113456554645512

    1. Initial program 57.8

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

    2. Taylor expanded in n around inf 56.8

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

    3. Simplified14.7

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

    4. Applied *-un-lft-identity_binary6414.7

      \[\leadsto 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{\color{blue}{1 \cdot i}} \]

    5. Applied times-frac_binary649.4

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{n}{1} \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

    6. Simplified9.4

      \[\leadsto 100 \cdot \left(\color{blue}{n} \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \]

    if 0.0216113456554645512 < i < 4.3408139321053756e272

    1. Initial program 32.0

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

    2. Taylor expanded in n around 0 21.0

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\frac{\log i \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \left(\frac{{n}^{2} \cdot \log i}{i} + \left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4} \cdot \left(\log i \cdot {\log n}^{2}\right)}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + 0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{4} \cdot \left({\log i}^{2} \cdot \log n\right)}{i} + \left(\frac{{n}^{4} \cdot \log n}{{i}^{2}} + \left(\frac{{n}^{2} \cdot \log n}{i} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + 0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log n}^{3}}{i}\right)\right)\right)\right)\right)\right)} \]

    if 4.3408139321053756e272 < i

    1. Initial program 33.0

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

    2. Simplified32.9

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -21315269.074093234:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.02161134565546455:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;i \leq 4.3408139321053756 \cdot 10^{+272}:\\ \;\;\;\;100 \cdot \left(\left(\frac{\log i \cdot {n}^{4}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{3} \cdot {\log i}^{2}}{i} + \left(\frac{\log i \cdot {n}^{2}}{i} + \left(\frac{{n}^{3}}{{i}^{2}} + \left(0.5 \cdot \frac{{n}^{4} \cdot \left(\log i \cdot {\log n}^{2}\right)}{i} + \left(0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log i}^{3}}{i} + 0.5 \cdot \frac{{n}^{3} \cdot {\log n}^{2}}{i}\right)\right)\right)\right)\right)\right) - \left(0.5 \cdot \frac{{n}^{4} \cdot \left({\log i}^{2} \cdot \log n\right)}{i} + \left(\frac{{n}^{4} \cdot \log n}{{i}^{2}} + \left(\frac{{n}^{2} \cdot \log n}{i} + \left(0.5 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + 0.16666666666666666 \cdot \frac{{n}^{4} \cdot {\log n}^{3}}{i}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(\frac{i}{n} + 1\right)}^{n}, -100\right)}{i}\\ \end{array} \]

Reproduce

herbie shell --seed 2022024 
(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))))