Compound Interest

Average Accuracy: 25.1% → 85.0%

Time: 28.9s

Precision: binary64

Cost: 46604

Math

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

↓

\[\begin{array}{l} t_0 := \log i - \log n\\ \mathbf{if}\;i \leq -7.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot \left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} + -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{-166}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right) + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 1.85 \cdot 10^{-22}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{t_0} + \mathsf{expm1}\left(n \cdot t_0\right)\right)}{\frac{i}{n}}\\ \end{array} \]

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 (- (log i) (log n))))
   (if (<= i -7.4e-12)
     (/ (* 100.0 (+ (exp (* n (log1p (/ i n)))) -1.0)) (/ i n))
     (if (<= i 2.1e-166)
       (* 100.0 (+ (* n (+ 1.0 (* i 0.5))) (* i -0.5)))
       (if (<= i 1.85e-22)
         (/ 100.0 (/ i (* n (expm1 i))))
         (/
          (* 100.0 (+ (* (/ (* n n) i) (pow (exp n) t_0)) (expm1 (* n t_0))))
          (/ 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 = log(i) - log(n);
	double tmp;
	if (i <= -7.4e-12) {
		tmp = (100.0 * (exp((n * log1p((i / n)))) + -1.0)) / (i / n);
	} else if (i <= 2.1e-166) {
		tmp = 100.0 * ((n * (1.0 + (i * 0.5))) + (i * -0.5));
	} else if (i <= 1.85e-22) {
		tmp = 100.0 / (i / (n * expm1(i)));
	} else {
		tmp = (100.0 * ((((n * n) / i) * pow(exp(n), t_0)) + expm1((n * t_0)))) / (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 = Math.log(i) - Math.log(n);
	double tmp;
	if (i <= -7.4e-12) {
		tmp = (100.0 * (Math.exp((n * Math.log1p((i / n)))) + -1.0)) / (i / n);
	} else if (i <= 2.1e-166) {
		tmp = 100.0 * ((n * (1.0 + (i * 0.5))) + (i * -0.5));
	} else if (i <= 1.85e-22) {
		tmp = 100.0 / (i / (n * Math.expm1(i)));
	} else {
		tmp = (100.0 * ((((n * n) / i) * Math.pow(Math.exp(n), t_0)) + Math.expm1((n * t_0)))) / (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 = math.log(i) - math.log(n)
	tmp = 0
	if i <= -7.4e-12:
		tmp = (100.0 * (math.exp((n * math.log1p((i / n)))) + -1.0)) / (i / n)
	elif i <= 2.1e-166:
		tmp = 100.0 * ((n * (1.0 + (i * 0.5))) + (i * -0.5))
	elif i <= 1.85e-22:
		tmp = 100.0 / (i / (n * math.expm1(i)))
	else:
		tmp = (100.0 * ((((n * n) / i) * math.pow(math.exp(n), t_0)) + math.expm1((n * t_0)))) / (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(log(i) - log(n))
	tmp = 0.0
	if (i <= -7.4e-12)
		tmp = Float64(Float64(100.0 * Float64(exp(Float64(n * log1p(Float64(i / n)))) + -1.0)) / Float64(i / n));
	elseif (i <= 2.1e-166)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(1.0 + Float64(i * 0.5))) + Float64(i * -0.5)));
	elseif (i <= 1.85e-22)
		tmp = Float64(100.0 / Float64(i / Float64(n * expm1(i))));
	else
		tmp = Float64(Float64(100.0 * Float64(Float64(Float64(Float64(n * n) / i) * (exp(n) ^ t_0)) + expm1(Float64(n * t_0)))) / 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[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -7.4e-12], N[(N[(100.0 * N[(N[Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.1e-166], N[(100.0 * N[(N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.85e-22], N[(100.0 / N[(i / N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(N[(N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision] * N[Power[N[Exp[n], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[(n * t$95$0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	25.1%
Target	25.8%
Herbie	85.0%

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

Derivation

1. Split input into 4 regimes

2. if i < -7.39999999999999997e-12

   1. Initial program 54.6%

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

   2. Simplified 54.6%

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

      [Start] 54.6	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 54.6	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 54.6	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 54.6	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Applied egg-rr 90.3%

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

      [Start] 54.6	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}{\frac{i}{n}} \]
      pow-to-exp [=>] 54.6	\[ \frac{100 \cdot \left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} + -1\right)}{\frac{i}{n}} \]
      *-commutative [=>] 54.6	\[ \frac{100 \cdot \left(e^{\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}} + -1\right)}{\frac{i}{n}} \]
      log1p-def [=>] 90.3	\[ \frac{100 \cdot \left(e^{n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}} + -1\right)}{\frac{i}{n}} \]

   if -7.39999999999999997e-12 < i < 2.0999999999999999e-166

   1. Initial program 7.1%

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

   2. Simplified 7.7%

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

      [Start] 7.1	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 7.7	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 7.7	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 7.7	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 7.7	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 7.7	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 7.7	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 7.7	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in i around 0 5.5%

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

   4. Simplified 5.5%

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

      [Start] 5.5	\[ 100 \cdot \left(n \cdot \frac{\left(1 + \left(i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + -1}{i}\right) \]
      unpow2 [=>] 5.5	\[ 100 \cdot \left(n \cdot \frac{\left(1 + \left(i + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + -1}{i}\right) \]
      associate-*r/ [=>] 5.5	\[ 100 \cdot \left(n \cdot \frac{\left(1 + \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) + -1}{i}\right) \]
      metadata-eval [=>] 5.5	\[ 100 \cdot \left(n \cdot \frac{\left(1 + \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) + -1}{i}\right) \]

   5. Taylor expanded in n around 0 89.6%

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

   if 2.0999999999999999e-166 < i < 1.85e-22

   1. Initial program 11.6%

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

   2. Simplified 11.9%

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

      [Start] 11.6	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 11.9	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 11.9	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 11.9	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 11.9	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 11.9	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 11.9	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 11.9	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 11.9%

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

   4. Simplified 79.7%

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

      [Start] 11.9	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      associate-/l* [=>] 11.9	\[ 100 \cdot \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \]
      expm1-def [=>] 79.7	\[ 100 \cdot \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \]

   5. Applied egg-rr 84.0%

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

      [Start] 79.7	\[ 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \]
      associate-*r/ [=>] 79.7	\[ \color{blue}{\frac{100 \cdot n}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]
      associate-/l* [=>] 79.5	\[ \color{blue}{\frac{100}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]
      associate-/l/ [=>] 84.0	\[ \frac{100}{\color{blue}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   if 1.85e-22 < i

   1. Initial program 46.2%

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

   2. Simplified 46.2%

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

      [Start] 46.2	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 46.2	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 46.2	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 46.2	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in i around inf 48.9%

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

   4. Simplified 63.1%

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

      [Start] 48.9	\[ \frac{100 \cdot \left(\left(\frac{{n}^{2} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)}}{i} + e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)}\right) - 1\right)}{\frac{i}{n}} \]
      associate--l+ [=>] 49.0	\[ \frac{100 \cdot \color{blue}{\left(\frac{{n}^{2} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)}}{i} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}}{\frac{i}{n}} \]
      associate-/l* [=>] 49.0	\[ \frac{100 \cdot \left(\color{blue}{\frac{{n}^{2}}{\frac{i}{e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)}}}} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      associate-/r/ [=>] 49.0	\[ \frac{100 \cdot \left(\color{blue}{\frac{{n}^{2}}{i} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)}} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      unpow2 [=>] 49.0	\[ \frac{100 \cdot \left(\frac{\color{blue}{n \cdot n}}{i} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      exp-prod [=>] 49.0	\[ \frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot \color{blue}{{\left(e^{n}\right)}^{\left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)}} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      mul-1-neg [=>] 49.0	\[ \frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\color{blue}{\left(-\log \left(\frac{1}{i}\right)\right)} + \log \left(\frac{1}{n}\right)\right)} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      log-rec [=>] 49.0	\[ \frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\left(-\color{blue}{\left(-\log i\right)}\right) + \log \left(\frac{1}{n}\right)\right)} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      remove-double-neg [=>] 49.0	\[ \frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\color{blue}{\log i} + \log \left(\frac{1}{n}\right)\right)} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      log-rec [=>] 49.0	\[ \frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\log i + \color{blue}{\left(-\log n\right)}\right)} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      unsub-neg [=>] 49.0	\[ \frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\color{blue}{\left(\log i - \log n\right)}} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)} - 1\right)\right)}{\frac{i}{n}} \]
      expm1-def [=>] 63.1	\[ \frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\log i - \log n\right)} + \color{blue}{\mathsf{expm1}\left(n \cdot \left(-1 \cdot \log \left(\frac{1}{i}\right) + \log \left(\frac{1}{n}\right)\right)\right)}\right)}{\frac{i}{n}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot \left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} + -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{-166}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right) + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 1.85 \cdot 10^{-22}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\log i - \log n\right)} + \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)\right)}{\frac{i}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.0%
Cost	20300

\[\begin{array}{l} \mathbf{if}\;i \leq -5.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot \left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} + -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right) + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log i - \log n\right)\right)}{\frac{i}{n}}\\ \end{array} \]

Alternative 2
Accuracy	84.7%
Cost	14284

\[\begin{array}{l} t_0 := \frac{\frac{i}{100}}{n}\\ \mathbf{if}\;i \leq -7.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot \left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} + -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 10^{-168}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right) + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 3.9 \cdot 10^{-22}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{\log i}{t_0} - \frac{\log n}{t_0}\right)\\ \end{array} \]

Alternative 3
Accuracy	83.3%
Cost	13900

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -3 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-198}:\\ \;\;\;\;n \cdot \frac{\left(\log i - \log n\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	84.7%
Cost	13900

\[\begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot \left(e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} + -1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-169}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right) + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)}{i}\\ \end{array} \]

Alternative 5
Accuracy	82.3%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;i \leq 5.4 \cdot 10^{-164}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(\left(\log i - \log n\right) \cdot \left(100 \cdot n\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	80.8%
Cost	7500

\[\begin{array}{l} \mathbf{if}\;i \leq 4.8 \cdot 10^{-161}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 3.85 \cdot 10^{+89}:\\ \;\;\;\;n \cdot \left(n \cdot \frac{\log \left(\frac{i}{n}\right)}{\frac{i}{100}}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{i} + \frac{n}{i} \cdot -100\\ \end{array} \]

Alternative 7
Accuracy	80.8%
Cost	7500

\[\begin{array}{l} \mathbf{if}\;i \leq 2.2 \cdot 10^{-166}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 7.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{100}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;i \leq 4.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{n}{i} \cdot \left(\log \left(\frac{i}{n}\right) \cdot \left(100 \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{i} + \frac{n}{i} \cdot -100\\ \end{array} \]

Alternative 8
Accuracy	82.2%
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -9.6 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-251}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \mathbf{elif}\;n \leq 100:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	82.2%
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-251}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \mathbf{elif}\;n \leq 100:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	70.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-187}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-251}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 11
Accuracy	70.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-187}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-253}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(-0.005 + i \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 12
Accuracy	69.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{-185} \lor \neg \left(n \leq 6.8 \cdot 10^{-251}\right):\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \end{array} \]

Alternative 13
Accuracy	69.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-187} \lor \neg \left(n \leq 5.4 \cdot 10^{-253}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \end{array} \]

Alternative 14
Accuracy	67.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+39} \lor \neg \left(i \leq 1.12 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]

Alternative 15
Accuracy	3.0%
Cost	192

\[i \cdot -50 \]

Alternative 16
Accuracy	56.3%
Cost	192

\[100 \cdot n \]

Reproduce

herbie shell --seed 2023137 
(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))))