Compound Interest

Average Error: 48.0 → 1.0

Time: 20.9s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 10^{-205}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_1 \leq 2000:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \end{array} \]

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 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 1e-205)
     (/ 100.0 (/ (/ i n) (expm1 (* n (log1p (/ i n))))))
     (if (<= t_1 2000.0)
       (* 100.0 (* n (/ t_0 i)))
       (/ 100.0 (/ (+ 1.0 (* i -0.5)) 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 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 1e-205) {
		tmp = 100.0 / ((i / n) / expm1((n * log1p((i / n)))));
	} else if (t_1 <= 2000.0) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = 100.0 / ((1.0 + (i * -0.5)) / 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 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 1e-205) {
		tmp = 100.0 / ((i / n) / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_1 <= 2000.0) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = 100.0 / ((1.0 + (i * -0.5)) / 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 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= 1e-205:
		tmp = 100.0 / ((i / n) / math.expm1((n * math.log1p((i / n)))))
	elif t_1 <= 2000.0:
		tmp = 100.0 * (n * (t_0 / i))
	else:
		tmp = 100.0 / ((1.0 + (i * -0.5)) / 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((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 1e-205)
		tmp = Float64(100.0 / Float64(Float64(i / n) / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_1 <= 2000.0)
		tmp = Float64(100.0 * Float64(n * Float64(t_0 / i)));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / 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[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-205], N[(100.0 / N[(N[(i / n), $MachinePrecision] / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(100.0 * N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	48.0
Target	47.4
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-205

   1. Initial program 46.5

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

   2. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   if 1e-205 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2e3

   1. Initial program 1.8

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

   2. Simplified 1.8

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

      [Start] 1.8	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 1.8	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 1.8	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 1.8	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 1.8	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 1.8	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 1.8	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 1.8	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   if 2e3 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 63.0

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

   2. Applied egg-rr 62.9

      \[\leadsto \color{blue}{\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   3. Taylor expanded in i around 0 1.1

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

   4. Simplified 1.1

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

      [Start] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)} \]
      associate-*r/ [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)} \]
      metadata-eval [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)} \]
      unpow2 [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)} \]
      associate-*r/ [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)} \]
      metadata-eval [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)} \]

   5. Taylor expanded in i around 0 1.1

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

   6. Simplified 1.1

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

      [Start] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)} \]
      associate-*r/ [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)} \]
      metadata-eval [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)} \]
      associate-*r/ [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - \frac{0.5}{n}\right)} \]
      metadata-eval [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - \frac{0.5}{n}\right)} \]
      unpow2 [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - \frac{0.5}{n}\right)} \]
      associate-/r* [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \left(\color{blue}{\frac{\frac{0.5}{n}}{n}} - \frac{0.5}{n}\right)} \]
      div-sub [<=] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \color{blue}{\frac{\frac{0.5}{n} - 0.5}{n}}} \]
      *-lft-identity [<=] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \color{blue}{\left(1 \cdot \frac{\frac{0.5}{n} - 0.5}{n}\right)}} \]
      *-commutative [<=] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \color{blue}{\left(\frac{\frac{0.5}{n} - 0.5}{n} \cdot 1\right)}} \]
      associate-*l/ [=>] 1.1	\[ \frac{100}{\frac{1}{n} + i \cdot \color{blue}{\frac{\left(\frac{0.5}{n} - 0.5\right) \cdot 1}{n}}} \]
      associate-*r/ [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \color{blue}{\frac{i \cdot \left(\left(\frac{0.5}{n} - 0.5\right) \cdot 1\right)}{n}}} \]
      *-commutative [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \color{blue}{\left(1 \cdot \left(\frac{0.5}{n} - 0.5\right)\right)}}{n}} \]
      sub-neg [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \left(1 \cdot \color{blue}{\left(\frac{0.5}{n} + \left(-0.5\right)\right)}\right)}{n}} \]
      metadata-eval [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \left(1 \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)}{n}} \]
      distribute-lft-in [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \color{blue}{\left(1 \cdot \frac{0.5}{n} + 1 \cdot -0.5\right)}}{n}} \]
      associate-*r/ [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \left(\color{blue}{\frac{1 \cdot 0.5}{n}} + 1 \cdot -0.5\right)}{n}} \]
      metadata-eval [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \left(\frac{\color{blue}{0.5}}{n} + 1 \cdot -0.5\right)}{n}} \]
      metadata-eval [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)}{n}} \]
      +-commutative [=>] 1.1	\[ \frac{100}{\frac{1}{n} + \frac{i \cdot \color{blue}{\left(-0.5 + \frac{0.5}{n}\right)}}{n}} \]

   7. Taylor expanded in n around inf 1.2

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

   8. Simplified 1.2

      \[\leadsto \frac{100}{\color{blue}{\frac{1 + i \cdot -0.5}{n}}} \]
      Proof

      [Start] 1.2	\[ \frac{100}{\frac{1 + -0.5 \cdot i}{n}} \]
      *-commutative [=>] 1.2	\[ \frac{100}{\frac{1 + \color{blue}{i \cdot -0.5}}{n}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-205}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2000:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 10^{-205}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 2000:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \end{array} \]

Alternative 2
Error	11.8
Cost	7692

\[\begin{array}{l} t_0 := -0.5 + \frac{0.5}{n}\\ \mathbf{if}\;i \leq -1.55 \cdot 10^{+63}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 20000000000:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot t_0}{n}}\\ \mathbf{elif}\;i \leq 1.05 \cdot 10^{+245}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \frac{t_0}{n}}\\ \end{array} \]

Alternative 3
Error	8.9
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -260 \lor \neg \left(n \leq 7400000000000\right):\\ \;\;\;\;\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot \left(-0.5 + \frac{0.5}{n}\right)}{n}}\\ \end{array} \]

Alternative 4
Error	8.8
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -260 \lor \neg \left(n \leq 5 \cdot 10^{-13}\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{i \cdot \left(-0.5 + \frac{0.5}{n}\right)}{n}}\\ \end{array} \]

Alternative 5
Error	17.4
Cost	1229

\[\begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{-0.5 + \frac{0.5}{n}}\\ \mathbf{elif}\;i \leq -1.6 \cdot 10^{-251} \lor \neg \left(i \leq 3.5 \cdot 10^{-200}\right):\\ \;\;\;\;\frac{100}{\frac{1}{n} + 0.5 \cdot \frac{i}{n \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	18.7
Cost	1097

\[\begin{array}{l} \mathbf{if}\;i \leq -0.17 \lor \neg \left(i \leq 0.14\right):\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{-0.5 + \frac{0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 + i \cdot 0.16666666666666666\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 7
Error	19.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;i \leq -3.9 \cdot 10^{+37} \lor \neg \left(i \leq 0.98\right):\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{-0.5 + \frac{0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 8
Error	16.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;n \leq -260:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + \frac{\frac{i}{n} \cdot 0.5}{n}}\\ \end{array} \]

Alternative 9
Error	16.0
Cost	960

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

Alternative 10
Error	20.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -3.35 \cdot 10^{-155} \lor \neg \left(n \leq 6.2 \cdot 10^{-228}\right):\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \left(n \cdot \frac{n}{i}\right)\\ \end{array} \]

Alternative 11
Error	20.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-150} \lor \neg \left(n \leq 5.2 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \left(n \cdot \frac{n}{i}\right)\\ \end{array} \]

Alternative 12
Error	20.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 0.8:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \left(n \cdot \frac{n}{i}\right)\\ \end{array} \]

Alternative 13
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -1.62:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 2.5:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \left(n \cdot \frac{n}{i}\right)\\ \end{array} \]

Alternative 14
Error	20.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -1.62:\\ \;\;\;\;\frac{100}{\frac{i}{n} \cdot -0.5}\\ \mathbf{elif}\;i \leq 7.8:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \left(n \cdot \frac{n}{i}\right)\\ \end{array} \]

Alternative 15
Error	21.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -2.8 \cdot 10^{+20}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1550000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16
Error	21.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 1550000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 17
Error	51.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023060 
(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))))