Average Error: 47.4 → 11.6
Time: 11.8s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -0.0001526379938816064:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{i}, 100, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.927304430705153 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, n \cdot \left(i \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\left(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left({\log i}^{2} + {\log n}^{2}\right) + 100 \cdot \left(n \cdot \log i + \frac{n \cdot n}{i}\right)\right) + -100 \cdot \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)\right)}{i}\\ \end{array} \]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -0.0001526379938816064:\\
\;\;\;\;\frac{\mathsf{fma}\left(e^{i}, 100, -100\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \leq 8.927304430705153 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(n, 100, n \cdot \left(i \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;n \cdot \frac{\left(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left({\log i}^{2} + {\log n}^{2}\right) + 100 \cdot \left(n \cdot \log i + \frac{n \cdot n}{i}\right)\right) + -100 \cdot \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)\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 -0.0001526379938816064)
   (/ (fma (exp i) 100.0 -100.0) (/ i n))
   (if (<= i 8.927304430705153e-7)
     (fma n 100.0 (* n (* i (fma 16.666666666666668 i 50.0))))
     (*
      n
      (/
       (+
        (+
         (* (* 50.0 (* n n)) (+ (pow (log i) 2.0) (pow (log n) 2.0)))
         (* 100.0 (+ (* n (log i)) (/ (* n n) i))))
        (* -100.0 (* (log n) (+ n (* (* n n) (log i))))))
       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 <= -0.0001526379938816064) {
		tmp = fma(exp(i), 100.0, -100.0) / (i / n);
	} else if (i <= 8.927304430705153e-7) {
		tmp = fma(n, 100.0, (n * (i * fma(16.666666666666668, i, 50.0))));
	} else {
		tmp = n * (((((50.0 * (n * n)) * (pow(log(i), 2.0) + pow(log(n), 2.0))) + (100.0 * ((n * log(i)) + ((n * n) / i)))) + (-100.0 * (log(n) * (n + ((n * n) * log(i)))))) / i);
	}
	return tmp;
}

Error

Bits error versus i

Bits error versus n

Target

Original47.4
Target47.3
Herbie11.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.52637993881606391e-4

    1. Initial program 28.2

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

    2. Simplified28.6

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

    3. Taylor expanded in n around inf 12.8

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

    4. Simplified11.8

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

    if -1.52637993881606391e-4 < i < 8.92730443070515327e-7

    1. Initial program 58.1

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

    2. Simplified57.8

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

    3. Taylor expanded in i around 0 9.0

      \[\leadsto \color{blue}{\left(16.666666666666668 \cdot \left(n \cdot {i}^{2}\right) + \left(100 \cdot n + \left(33.333333333333336 \cdot \frac{{i}^{2}}{n} + 50 \cdot \left(n \cdot i\right)\right)\right)\right) - \left(50 \cdot i + 50 \cdot {i}^{2}\right)} \]

    4. Simplified9.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(16.666666666666668, \left(i \cdot i\right) \cdot n, \mathsf{fma}\left(100, n, \mathsf{fma}\left(33.333333333333336, \frac{i \cdot i}{n}, 50 \cdot \left(i \cdot n\right)\right)\right)\right) - 50 \cdot \left(i + i \cdot i\right)} \]

    5. Taylor expanded in i around 0 9.0

      \[\leadsto \color{blue}{\left(16.666666666666668 \cdot \left(n \cdot {i}^{2}\right) + \left(100 \cdot n + \left(33.333333333333336 \cdot \frac{{i}^{2}}{n} + 50 \cdot \left(n \cdot i\right)\right)\right)\right) - \left(50 \cdot i + 50 \cdot {i}^{2}\right)} \]

    6. Simplified8.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(n, 100, i \cdot \left(i \cdot \left(\mathsf{fma}\left(n, 16.666666666666668, \frac{33.333333333333336}{n}\right) - 50\right) + \mathsf{fma}\left(n, 50, -50\right)\right)\right)} \]

    7. Taylor expanded in n around inf 9.1

      \[\leadsto \mathsf{fma}\left(n, 100, \color{blue}{n \cdot \left(i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)}\right) \]

    8. Simplified9.1

      \[\leadsto \mathsf{fma}\left(n, 100, \color{blue}{n \cdot \left(i \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right)\right)}\right) \]

    if 8.92730443070515327e-7 < i

    1. Initial program 33.6

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

    2. Simplified33.5

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

    3. Taylor expanded in n around 0 22.2

      \[\leadsto n \cdot \frac{\color{blue}{\left(50 \cdot \left({n}^{2} \cdot {\log i}^{2}\right) + \left(50 \cdot \left({n}^{2} \cdot {\log n}^{2}\right) + \left(100 \cdot \left(\log i \cdot n\right) + 100 \cdot \frac{{n}^{2}}{i}\right)\right)\right) - \left(100 \cdot \left({n}^{2} \cdot \left(\log i \cdot \log n\right)\right) + 100 \cdot \left(n \cdot \log n\right)\right)}}{i} \]

    4. Simplified22.1

      \[\leadsto n \cdot \frac{\color{blue}{\left(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left({\log i}^{2} + {\log n}^{2}\right) + 100 \cdot \left(n \cdot \log i + \frac{n \cdot n}{i}\right)\right) + -100 \cdot \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)\right)}}{i} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0001526379938816064:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{i}, 100, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.927304430705153 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, n \cdot \left(i \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\left(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left({\log i}^{2} + {\log n}^{2}\right) + 100 \cdot \left(n \cdot \log i + \frac{n \cdot n}{i}\right)\right) + -100 \cdot \left(\log n \cdot \left(n + \left(n \cdot n\right) \cdot \log i\right)\right)}{i}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))))