Average Error: 47.8 → 11.5
Time: 11.3s
Precision: binary64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq -1.2241496397660598 \cdot 10^{-10}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 0.8920348598017706:\\ \;\;\;\;\mathsf{fma}\left(16.666666666666668, n \cdot \left(i \cdot i\right), \mathsf{fma}\left(n, 100, \mathsf{fma}\left(\frac{i \cdot i}{n}, 33.333333333333336, 50 \cdot \left(i \cdot n\right)\right)\right)\right) - 50 \cdot \left(i + i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(\frac{i}{n} + 1\right)}^{n}, -100\right)}{i}\\ \end{array} \]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \leq -1.2241496397660598 \cdot 10^{-10}:\\
\;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}{i}\\

\mathbf{elif}\;i \leq 0.8920348598017706:\\
\;\;\;\;\mathsf{fma}\left(16.666666666666668, n \cdot \left(i \cdot i\right), \mathsf{fma}\left(n, 100, \mathsf{fma}\left(\frac{i \cdot i}{n}, 33.333333333333336, 50 \cdot \left(i \cdot n\right)\right)\right)\right) - 50 \cdot \left(i + i \cdot i\right)\\

\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 -1.2241496397660598e-10)
   (* n (/ (fma 100.0 (exp (* n (log1p (/ i n)))) -100.0) i))
   (if (<= i 0.8920348598017706)
     (-
      (fma
       16.666666666666668
       (* n (* i i))
       (fma n 100.0 (fma (/ (* i i) n) 33.333333333333336 (* 50.0 (* i n)))))
      (* 50.0 (+ i (* i 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 <= -1.2241496397660598e-10) {
		tmp = n * (fma(100.0, exp(n * log1p(i / n)), -100.0) / i);
	} else if (i <= 0.8920348598017706) {
		tmp = fma(16.666666666666668, (n * (i * i)), fma(n, 100.0, fma(((i * i) / n), 33.333333333333336, (50.0 * (i * n))))) - (50.0 * (i + (i * 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

Original47.8
Target47.8
Herbie11.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.2241496397660598e-10

    1. Initial program 27.8

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

    2. Simplified28.6

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

    3. Applied pow-to-exp_binary6428.6

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

    4. Simplified6.9

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

    if -1.2241496397660598e-10 < i < 0.892034859801770552

    1. Initial program 58.4

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

    2. Simplified58.1

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

    3. Taylor expanded in i around 0 8.8

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

    4. Simplified8.8

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

    if 0.892034859801770552 < i

    1. Initial program 32.7

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

    2. Simplified32.7

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.2241496397660598 \cdot 10^{-10}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, -100\right)}{i}\\ \mathbf{elif}\;i \leq 0.8920348598017706:\\ \;\;\;\;\mathsf{fma}\left(16.666666666666668, n \cdot \left(i \cdot i\right), \mathsf{fma}\left(n, 100, \mathsf{fma}\left(\frac{i \cdot i}{n}, 33.333333333333336, 50 \cdot \left(i \cdot n\right)\right)\right)\right) - 50 \cdot \left(i + i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(\frac{i}{n} + 1\right)}^{n}, -100\right)}{i}\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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))))