Compound Interest

Average Error: 47.8 → 10.8

Time: 26.4s

Precision: binary64

Cost: 13900

Math

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

↓

\[\begin{array}{l} t_0 := n \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{if}\;i \leq -5.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{100}{\frac{i}{t_0}}\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 500:\\ \;\;\;\;t_0 \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \left(\log i - \log n\right)\right)}{\frac{i}{n}}\\ \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

↓

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (expm1 i))))
   (if (<= i -5.1e-225)
     (/ 100.0 (/ i t_0))
     (if (<= i 8.5e-141)
       (* 100.0 (+ n (* i (* n (+ 0.5 (/ -0.5 n))))))
       (if (<= i 500.0)
         (* t_0 (/ 100.0 i))
         (/ (* n (* 100.0 (- (log i) (log n)))) (/ i n)))))))

C

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 t_0 = n * expm1(i);
	double tmp;
	if (i <= -5.1e-225) {
		tmp = 100.0 / (i / t_0);
	} else if (i <= 8.5e-141) {
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))));
	} else if (i <= 500.0) {
		tmp = t_0 * (100.0 / i);
	} else {
		tmp = (n * (100.0 * (log(i) - log(n)))) / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

↓

public static double code(double i, double n) {
	double t_0 = n * Math.expm1(i);
	double tmp;
	if (i <= -5.1e-225) {
		tmp = 100.0 / (i / t_0);
	} else if (i <= 8.5e-141) {
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))));
	} else if (i <= 500.0) {
		tmp = t_0 * (100.0 / i);
	} else {
		tmp = (n * (100.0 * (Math.log(i) - Math.log(n)))) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

↓

def code(i, n):
	t_0 = n * math.expm1(i)
	tmp = 0
	if i <= -5.1e-225:
		tmp = 100.0 / (i / t_0)
	elif i <= 8.5e-141:
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))))
	elif i <= 500.0:
		tmp = t_0 * (100.0 / i)
	else:
		tmp = (n * (100.0 * (math.log(i) - math.log(n)))) / (i / n)
	return tmp

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

↓

function code(i, n)
	t_0 = Float64(n * expm1(i))
	tmp = 0.0
	if (i <= -5.1e-225)
		tmp = Float64(100.0 / Float64(i / t_0));
	elseif (i <= 8.5e-141)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(0.5 + Float64(-0.5 / n))))));
	elseif (i <= 500.0)
		tmp = Float64(t_0 * Float64(100.0 / i));
	else
		tmp = Float64(Float64(n * Float64(100.0 * Float64(log(i) - log(n)))) / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[i_, n_] := Block[{t$95$0 = N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5.1e-225], N[(100.0 / N[(i / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.5e-141], N[(100.0 * N[(n + N[(i * N[(n * N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 500.0], N[(t$95$0 * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(100.0 * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	47.8
Target	47.5
Herbie	10.8

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

Derivation

1. Split input into 4 regimes

2. if i < -5.0999999999999999e-225

   1. Initial program 42.0

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

   2. Simplified 42.2

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

      [Start] 42.0	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 42.2	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 42.2	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 42.2	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 42.2	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 42.2	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 42.2	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 42.2	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 33.9

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

   4. Simplified 11.7

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

      [Start] 33.9	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      associate-*r/ [=>] 34.0	\[ \color{blue}{\frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i}} \]
      associate-/l* [=>] 33.5	\[ \color{blue}{\frac{100}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]
      expm1-def [=>] 11.7	\[ \frac{100}{\frac{i}{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}} \]

   if -5.0999999999999999e-225 < i < 8.50000000000000021e-141

   1. Initial program 60.8

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

   2. Simplified 60.3

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

      [Start] 60.8	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 60.3	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 60.3	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 60.3	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 60.3	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 60.3	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 60.3	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 60.3	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in i around 0 4.2

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

   4. Simplified 4.2

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

      [Start] 4.2	\[ 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n \]
      distribute-lft-out [=>] 4.2	\[ \color{blue}{100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n\right)} \]
      +-commutative [<=] 4.2	\[ 100 \cdot \color{blue}{\left(n + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
      *-commutative [=>] 4.2	\[ \color{blue}{\left(n + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \cdot 100} \]
      associate-*r* [=>] 4.3	\[ \left(n + \color{blue}{\left(n \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right) \cdot 100 \]
      *-commutative [=>] 4.3	\[ \left(n + \color{blue}{\left(i \cdot n\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100 \]
      associate-*l* [=>] 4.2	\[ \left(n + \color{blue}{i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \cdot 100 \]
      cancel-sign-sub-inv [=>] 4.2	\[ \left(n + i \cdot \left(n \cdot \color{blue}{\left(0.5 + \left(-0.5\right) \cdot \frac{1}{n}\right)}\right)\right) \cdot 100 \]
      metadata-eval [=>] 4.2	\[ \left(n + i \cdot \left(n \cdot \left(0.5 + \color{blue}{-0.5} \cdot \frac{1}{n}\right)\right)\right) \cdot 100 \]
      associate-*r/ [=>] 4.2	\[ \left(n + i \cdot \left(n \cdot \left(0.5 + \color{blue}{\frac{-0.5 \cdot 1}{n}}\right)\right)\right) \cdot 100 \]
      metadata-eval [=>] 4.2	\[ \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{\color{blue}{-0.5}}{n}\right)\right)\right) \cdot 100 \]

   if 8.50000000000000021e-141 < i < 500

   1. Initial program 54.3

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

   2. Simplified 54.2

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

      [Start] 54.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 54.2	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 54.2	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 54.2	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 54.2	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 54.2	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 54.2	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 54.2	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 52.8

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

   4. Simplified 13.0

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

      [Start] 52.8	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      associate-*r/ [=>] 52.8	\[ \color{blue}{\frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i}} \]
      associate-/l* [=>] 52.8	\[ \color{blue}{\frac{100}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]
      expm1-def [=>] 13.0	\[ \frac{100}{\frac{i}{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}} \]

   5. Applied egg-rr 13.0

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

   if 500 < i

   1. Initial program 33.3

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

   2. Simplified 33.3

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

      [Start] 33.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 33.3	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 33.3	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 33.3	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in n around 0 19.6

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

   4. Simplified 19.6

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

      [Start] 19.6	\[ \frac{100 \cdot \left(n \cdot \left(-1 \cdot \log n + \log i\right)\right)}{\frac{i}{n}} \]
      associate-*r* [=>] 19.6	\[ \frac{\color{blue}{\left(100 \cdot n\right) \cdot \left(-1 \cdot \log n + \log i\right)}}{\frac{i}{n}} \]
      *-commutative [=>] 19.6	\[ \frac{\color{blue}{\left(n \cdot 100\right)} \cdot \left(-1 \cdot \log n + \log i\right)}{\frac{i}{n}} \]
      associate-*l* [=>] 19.6	\[ \frac{\color{blue}{n \cdot \left(100 \cdot \left(-1 \cdot \log n + \log i\right)\right)}}{\frac{i}{n}} \]
      +-commutative [=>] 19.6	\[ \frac{n \cdot \left(100 \cdot \color{blue}{\left(\log i + -1 \cdot \log n\right)}\right)}{\frac{i}{n}} \]
      mul-1-neg [=>] 19.6	\[ \frac{n \cdot \left(100 \cdot \left(\log i + \color{blue}{\left(-\log n\right)}\right)\right)}{\frac{i}{n}} \]
      unsub-neg [=>] 19.6	\[ \frac{n \cdot \left(100 \cdot \color{blue}{\left(\log i - \log n\right)}\right)}{\frac{i}{n}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 500:\\ \;\;\;\;\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \left(\log i - \log n\right)\right)}{\frac{i}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	13900

\[\begin{array}{l} t_0 := n \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{if}\;i \leq -7 \cdot 10^{-225}:\\ \;\;\;\;\frac{100}{\frac{i}{t_0}}\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 500:\\ \;\;\;\;t_0 \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot \left(\log i - \log n\right)}{i}\right)\\ \end{array} \]

Alternative 2
Error	12.1
Cost	7244

\[\begin{array}{l} t_0 := \left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{if}\;i \leq -7 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 550:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	11.9
Cost	7244

\[\begin{array}{l} t_0 := n \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{if}\;i \leq -5.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{100}{\frac{i}{t_0}}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 540:\\ \;\;\;\;t_0 \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	12.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 6.5 \cdot 10^{-140}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 200:\\ \;\;\;\;\frac{100}{\frac{i}{i \cdot \left(n + i \cdot \left(n \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	19.6
Cost	976

\[\begin{array}{l} \mathbf{if}\;i \leq -0.41:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{-141}:\\ \;\;\;\;i \cdot \left(n \cdot 50 + -50\right) + 100 \cdot n\\ \mathbf{elif}\;i \leq 1.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;i \leq 205:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	19.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-272} \lor \neg \left(n \leq 2.6 \cdot 10^{-184}\right):\\ \;\;\;\;\frac{100}{\frac{1}{n} + -0.5 \cdot \frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	19.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;i \leq -1.55 \cdot 10^{+14}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{-140}:\\ \;\;\;\;100 \cdot n + i \cdot -50\\ \mathbf{elif}\;i \leq 200:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	19.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -0.08:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 200:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	20.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -2.15 \cdot 10^{+16}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 200:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	50.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023038 
(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))))