Compound Interest

Average Accuracy: 25.7% → 80.8%

Time: 28.6s

Precision: binary64

Cost: 46984

Math

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

↓

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ t_1 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_2 := \log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\\ \mathbf{if}\;n \leq -2.15 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{t_2} + \mathsf{expm1}\left(n \cdot t_2\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* 100.0 (/ n (+ 1.0 (* i (+ -0.5 (* i 0.08333333333333333)))))))
        (t_1 (* 100.0 (/ n (/ i (expm1 i)))))
        (t_2 (- (log (/ -1.0 n)) (log (/ -1.0 i)))))
   (if (<= n -2.15e-29)
     t_1
     (if (<= n -1.35e-64)
       (/
        (* 100.0 (+ (* (/ (* n n) i) (pow (exp n) t_2)) (expm1 (* n t_2))))
        (/ i n))
       (if (<= n -1e-201)
         t_0
         (if (<= n 5.5e-300)
           (/ (* 100.0 (expm1 (* n (log (/ i n))))) (/ i n))
           (if (<= n 5e+80) t_0 t_1)))))))

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 = 100.0 * (n / (1.0 + (i * (-0.5 + (i * 0.08333333333333333)))));
	double t_1 = 100.0 * (n / (i / expm1(i)));
	double t_2 = log((-1.0 / n)) - log((-1.0 / i));
	double tmp;
	if (n <= -2.15e-29) {
		tmp = t_1;
	} else if (n <= -1.35e-64) {
		tmp = (100.0 * ((((n * n) / i) * pow(exp(n), t_2)) + expm1((n * t_2)))) / (i / n);
	} else if (n <= -1e-201) {
		tmp = t_0;
	} else if (n <= 5.5e-300) {
		tmp = (100.0 * expm1((n * log((i / n))))) / (i / n);
	} else if (n <= 5e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = 100.0 * (n / (1.0 + (i * (-0.5 + (i * 0.08333333333333333)))));
	double t_1 = 100.0 * (n / (i / Math.expm1(i)));
	double t_2 = Math.log((-1.0 / n)) - Math.log((-1.0 / i));
	double tmp;
	if (n <= -2.15e-29) {
		tmp = t_1;
	} else if (n <= -1.35e-64) {
		tmp = (100.0 * ((((n * n) / i) * Math.pow(Math.exp(n), t_2)) + Math.expm1((n * t_2)))) / (i / n);
	} else if (n <= -1e-201) {
		tmp = t_0;
	} else if (n <= 5.5e-300) {
		tmp = (100.0 * Math.expm1((n * Math.log((i / n))))) / (i / n);
	} else if (n <= 5e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = 100.0 * (n / (1.0 + (i * (-0.5 + (i * 0.08333333333333333)))))
	t_1 = 100.0 * (n / (i / math.expm1(i)))
	t_2 = math.log((-1.0 / n)) - math.log((-1.0 / i))
	tmp = 0
	if n <= -2.15e-29:
		tmp = t_1
	elif n <= -1.35e-64:
		tmp = (100.0 * ((((n * n) / i) * math.pow(math.exp(n), t_2)) + math.expm1((n * t_2)))) / (i / n)
	elif n <= -1e-201:
		tmp = t_0
	elif n <= 5.5e-300:
		tmp = (100.0 * math.expm1((n * math.log((i / n))))) / (i / n)
	elif n <= 5e+80:
		tmp = t_0
	else:
		tmp = t_1
	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(100.0 * Float64(n / Float64(1.0 + Float64(i * Float64(-0.5 + Float64(i * 0.08333333333333333))))))
	t_1 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	t_2 = Float64(log(Float64(-1.0 / n)) - log(Float64(-1.0 / i)))
	tmp = 0.0
	if (n <= -2.15e-29)
		tmp = t_1;
	elseif (n <= -1.35e-64)
		tmp = Float64(Float64(100.0 * Float64(Float64(Float64(Float64(n * n) / i) * (exp(n) ^ t_2)) + expm1(Float64(n * t_2)))) / Float64(i / n));
	elseif (n <= -1e-201)
		tmp = t_0;
	elseif (n <= 5.5e-300)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log(Float64(i / n))))) / Float64(i / n));
	elseif (n <= 5e+80)
		tmp = t_0;
	else
		tmp = t_1;
	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[(100.0 * N[(n / N[(1.0 + N[(i * N[(-0.5 + N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.15e-29], t$95$1, If[LessEqual[n, -1.35e-64], N[(N[(100.0 * N[(N[(N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision] * N[Power[N[Exp[n], $MachinePrecision], t$95$2], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[(n * t$95$2), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1e-201], t$95$0, If[LessEqual[n, 5.5e-300], N[(N[(100.0 * N[(Exp[N[(n * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e+80], t$95$0, t$95$1]]]]]]]]

Target

Original	25.7%
Target	26.4%
Herbie	80.8%

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

Derivation

1. Split input into 4 regimes

2. if n < -2.1499999999999999e-29 or 4.99999999999999961e80 < n

   1. Initial program 21.0%

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

   2. Simplified 21.6%

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

      [Start] 21.0	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 21.6	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 21.6	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 21.6	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 21.6	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 21.6	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 21.6	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 21.6	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 29.9%

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

   4. Simplified 89.9%

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

      [Start] 29.9	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      associate-/l* [=>] 29.9	\[ 100 \cdot \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \]
      expm1-def [=>] 89.9	\[ 100 \cdot \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \]

   if -2.1499999999999999e-29 < n < -1.34999999999999993e-64

   1. Initial program 11.1%

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

   2. Simplified 11.1%

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

      [Start] 11.1	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 11.1	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 11.1	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 11.1	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in i around -inf 9.8%

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

   4. Simplified 38.6%

      \[\leadsto \frac{100 \cdot \color{blue}{\left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)} + \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)\right)}}{\frac{i}{n}} \]
      Proof

      [Start] 9.8	\[ \frac{100 \cdot \left(\left(\frac{{n}^{2} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)}}{i} + e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)}\right) - 1\right)}{\frac{i}{n}} \]
      associate--l+ [=>] 10.6	\[ \frac{100 \cdot \color{blue}{\left(\frac{{n}^{2} \cdot e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)}}{i} + \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)} - 1\right)\right)}}{\frac{i}{n}} \]

   if -1.34999999999999993e-64 < n < -9.99999999999999946e-202 or 5.4999999999999999e-300 < n < 4.99999999999999961e80

   1. Initial program 24.3%

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

   2. Simplified 24.2%

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

      [Start] 24.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 24.2	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 24.2	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 24.2	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 24.2	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 24.2	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 24.2	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 24.2	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 14.3%

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

   4. Simplified 54.5%

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

      [Start] 14.3	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      associate-/l* [=>] 14.3	\[ 100 \cdot \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \]
      expm1-def [=>] 54.5	\[ 100 \cdot \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \]

   5. Taylor expanded in i around 0 71.5%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \]

   6. Simplified 71.5%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + i \cdot \left(-0.5 + 0.08333333333333333 \cdot i\right)}} \]
      Proof

      [Start] 71.5	\[ 100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)} \]
      +-commutative [=>] 71.5	\[ 100 \cdot \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
      unpow2 [=>] 71.5	\[ 100 \cdot \frac{n}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot \color{blue}{\left(i \cdot i\right)}\right)} \]
      associate-*r* [=>] 71.5	\[ 100 \cdot \frac{n}{1 + \left(-0.5 \cdot i + \color{blue}{\left(0.08333333333333333 \cdot i\right) \cdot i}\right)} \]
      distribute-rgt-out [=>] 71.5	\[ 100 \cdot \frac{n}{1 + \color{blue}{i \cdot \left(-0.5 + 0.08333333333333333 \cdot i\right)}} \]

   if -9.99999999999999946e-202 < n < 5.4999999999999999e-300

   1. Initial program 73.8%

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

   2. Simplified 73.8%

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

      [Start] 73.8	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 73.8	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 73.8	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 73.8	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Taylor expanded in i around -inf 69.5%

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

   4. Simplified 76.7%

      \[\leadsto \frac{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}}{\frac{i}{n}} \]
      Proof

      [Start] 69.5	\[ \frac{100 \cdot \left(e^{n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)} - 1\right)}{\frac{i}{n}} \]
      expm1-def [=>] 76.7	\[ \frac{100 \cdot \color{blue}{\mathsf{expm1}\left(n \cdot \left(-1 \cdot \log \left(\frac{-1}{i}\right) + \log \left(-\frac{1}{n}\right)\right)\right)}}{\frac{i}{n}} \]
      +-commutative [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\left(\log \left(-\frac{1}{n}\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)}\right)}{\frac{i}{n}} \]
      mul-1-neg [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(-\frac{1}{n}\right) + \color{blue}{\left(-\log \left(\frac{-1}{i}\right)\right)}\right)\right)}{\frac{i}{n}} \]
      unsub-neg [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\left(\log \left(-\frac{1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)}\right)}{\frac{i}{n}} \]
      distribute-neg-frac [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \color{blue}{\left(\frac{-1}{n}\right)} - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}} \]
      metadata-eval [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{\color{blue}{-1}}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}} \]

   5. Applied egg-rr 80.5%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\left(0 + \log \left(\frac{1}{n} \cdot i\right)\right)}\right)}{\frac{i}{n}} \]
      Proof

      [Start] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}} \]
      *-un-lft-identity [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \color{blue}{\left(1 \cdot \frac{-1}{n}\right)} - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}} \]
      log-prod [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\color{blue}{\left(\log 1 + \log \left(\frac{-1}{n}\right)\right)} - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}} \]
      associate--l+ [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\left(\log 1 + \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}\right)}{\frac{i}{n}} \]
      metadata-eval [=>] 76.7	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\color{blue}{0} + \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)\right)}{\frac{i}{n}} \]
      diff-log [=>] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(0 + \color{blue}{\log \left(\frac{\frac{-1}{n}}{\frac{-1}{i}}\right)}\right)\right)}{\frac{i}{n}} \]
      associate-/r/ [=>] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(0 + \log \color{blue}{\left(\frac{\frac{-1}{n}}{-1} \cdot i\right)}\right)\right)}{\frac{i}{n}} \]
      associate-/l/ [=>] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(0 + \log \left(\color{blue}{\frac{-1}{-1 \cdot n}} \cdot i\right)\right)\right)}{\frac{i}{n}} \]
      metadata-eval [<=] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(0 + \log \left(\frac{\color{blue}{-1}}{-1 \cdot n} \cdot i\right)\right)\right)}{\frac{i}{n}} \]
      neg-mul-1 [<=] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(0 + \log \left(\frac{-1}{\color{blue}{-n}} \cdot i\right)\right)\right)}{\frac{i}{n}} \]
      frac-2neg [<=] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(0 + \log \left(\color{blue}{\frac{1}{n}} \cdot i\right)\right)\right)}{\frac{i}{n}} \]

   6. Simplified 80.5%

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

      [Start] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(0 + \log \left(\frac{1}{n} \cdot i\right)\right)\right)}{\frac{i}{n}} \]
      +-lft-identity [=>] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\log \left(\frac{1}{n} \cdot i\right)}\right)}{\frac{i}{n}} \]
      associate-*l/ [=>] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \log \color{blue}{\left(\frac{1 \cdot i}{n}\right)}\right)}{\frac{i}{n}} \]
      *-lft-identity [=>] 80.5	\[ \frac{100 \cdot \mathsf{expm1}\left(n \cdot \log \left(\frac{\color{blue}{i}}{n}\right)\right)}{\frac{i}{n}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{-29}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{100 \cdot \left(\frac{n \cdot n}{i} \cdot {\left(e^{n}\right)}^{\left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)} + \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+80}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.8%
Cost	20424

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_1 := 100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{if}\;n \leq -4.9 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq -8.8 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	80.6%
Cost	14032

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_1 := 100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ t_2 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.18 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq -7.8 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-300}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	80.7%
Cost	14032

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)\\ t_1 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_2 := 100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -1.4 \cdot 10^{-64}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{i}\right)\\ \mathbf{elif}\;n \leq -2.8 \cdot 10^{-204}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{100 \cdot t_0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	81.8%
Cost	7508

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_1 := 100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{if}\;n \leq -4.9 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.4 \cdot 10^{-64}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \log \left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-232}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	81.8%
Cost	7508

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_1 := 100 \cdot \frac{n}{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-64}:\\ \;\;\;\;\frac{100}{\frac{i}{\left(n \cdot n\right) \cdot \log \left(\frac{i}{n}\right)}}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	80.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{-42} \lor \neg \left(n \leq 5 \cdot 10^{+80}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}{n \cdot 100}}\\ \end{array} \]

Alternative 7
Accuracy	69.0%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;n \leq -3.9 \cdot 10^{-41}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot \left(-0.5 + i \cdot 0.08333333333333333\right)}{n \cdot 100}}\\ \end{array} \]

Alternative 8
Accuracy	69.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-208} \lor \neg \left(n \leq 8.8 \cdot 10^{-233}\right):\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternative 9
Accuracy	69.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-207} \lor \neg \left(n \leq 8.8 \cdot 10^{-233}\right):\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 10
Accuracy	69.7%
Cost	840

\[\begin{array}{l} t_0 := 1 + i \cdot -0.5\\ \mathbf{if}\;n \leq -7.2 \cdot 10^{-208}:\\ \;\;\;\;\frac{1}{\frac{t_0}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{t_0}\\ \end{array} \]

Alternative 11
Accuracy	66.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.6 \lor \neg \left(i \leq 4.15 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 12
Accuracy	66.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -1.62 \cdot 10^{-93}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+25}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]

Alternative 13
Accuracy	66.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 5.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 14
Accuracy	3.0%
Cost	192

\[i \cdot -50 \]

Alternative 15
Accuracy	55.7%
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023131 
(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))))