Compound Interest

Average Error: 47.3 → 7.4

Time: 25.2s

Precision: binary64

Cost: 14032

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\\ \mathbf{if}\;i \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{t_0}{\frac{i \cdot 0.01}{n}}\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{elif}\;i \leq 1000:\\ \;\;\;\;t_0 \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)}{\frac{i}{n}}\\ \end{array} \]

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 (expm1 (* n (log1p (/ i n))))))
   (if (<= i -3.6e-63)
     (/ t_0 (/ (* i 0.01) n))
     (if (<= i 1.656792211520858e-238)
       (* 100.0 (* n (+ 1.0 (* i (/ -0.5 n)))))
       (if (<= i 9.2e-114)
         (/ (* n (* i 100.0)) i)
         (if (<= i 1000.0)
           (* t_0 (* 100.0 (/ n i)))
           (/ (* 100.0 (* n (- (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 = expm1((n * log1p((i / n))));
	double tmp;
	if (i <= -3.6e-63) {
		tmp = t_0 / ((i * 0.01) / n);
	} else if (i <= 1.656792211520858e-238) {
		tmp = 100.0 * (n * (1.0 + (i * (-0.5 / n))));
	} else if (i <= 9.2e-114) {
		tmp = (n * (i * 100.0)) / i;
	} else if (i <= 1000.0) {
		tmp = t_0 * (100.0 * (n / i));
	} else {
		tmp = (100.0 * (n * (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 = Math.expm1((n * Math.log1p((i / n))));
	double tmp;
	if (i <= -3.6e-63) {
		tmp = t_0 / ((i * 0.01) / n);
	} else if (i <= 1.656792211520858e-238) {
		tmp = 100.0 * (n * (1.0 + (i * (-0.5 / n))));
	} else if (i <= 9.2e-114) {
		tmp = (n * (i * 100.0)) / i;
	} else if (i <= 1000.0) {
		tmp = t_0 * (100.0 * (n / i));
	} else {
		tmp = (100.0 * (n * (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 = math.expm1((n * math.log1p((i / n))))
	tmp = 0
	if i <= -3.6e-63:
		tmp = t_0 / ((i * 0.01) / n)
	elif i <= 1.656792211520858e-238:
		tmp = 100.0 * (n * (1.0 + (i * (-0.5 / n))))
	elif i <= 9.2e-114:
		tmp = (n * (i * 100.0)) / i
	elif i <= 1000.0:
		tmp = t_0 * (100.0 * (n / i))
	else:
		tmp = (100.0 * (n * (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 = expm1(Float64(n * log1p(Float64(i / n))))
	tmp = 0.0
	if (i <= -3.6e-63)
		tmp = Float64(t_0 / Float64(Float64(i * 0.01) / n));
	elseif (i <= 1.656792211520858e-238)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(-0.5 / n)))));
	elseif (i <= 9.2e-114)
		tmp = Float64(Float64(n * Float64(i * 100.0)) / i);
	elseif (i <= 1000.0)
		tmp = Float64(t_0 * Float64(100.0 * Float64(n / i)));
	else
		tmp = Float64(Float64(100.0 * Float64(n * 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[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]}, If[LessEqual[i, -3.6e-63], N[(t$95$0 / N[(N[(i * 0.01), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.656792211520858e-238], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 9.2e-114], N[(N[(n * N[(i * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 1000.0], N[(t$95$0 * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	47.3
Target	47.2
Herbie	7.4

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

Derivation

1. Split input into 5 regimes

2. if i < -3.60000000000000008e-63

   1. Initial program 32.9

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 1.5

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

   4. Applied egg-rr 1.2

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot 0.01}{n}}} \]

   if -3.60000000000000008e-63 < i < 1.656792211520858e-238

   1. Initial program 59.8

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

   2. Taylor expanded in i around 0 6.2

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

   3. Applied egg-rr 6.2

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

   4. Taylor expanded in n around 0 6.2

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

   5. Simplified 6.2

      \[\leadsto 100 \cdot \left(n \cdot \left(1 + \color{blue}{i \cdot \frac{-0.5}{n}}\right)\right) \]
      Proof
      (*.f64 i (/.f64 -1/2 n)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 i -1/2) n)): 31 points increase in error, 24 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 i)) n): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 i n))): 2 points increase in error, 0 points decrease in error

   if 1.656792211520858e-238 < i < 9.1999999999999997e-114

   1. Initial program 58.7

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

   2. Simplified 58.7

      \[\leadsto \color{blue}{\frac{n}{i} \cdot \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)} \]
      Proof
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) -100)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) (Rewrite<= metadata-eval (*.f64 -1 100)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (fma.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n) (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) 100))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 100 (pow.f64 (+.f64 1 (/.f64 i n)) n)) (*.f64 (neg.f64 1) 100)))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 n i) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 100)) (*.f64 (neg.f64 1) 100))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (Rewrite<= distribute-rgt-in_binary64 (*.f64 100 (+.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) (neg.f64 1))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 n i) (*.f64 100 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 n (/.f64 i (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))))): 3 points increase in error, 48 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 n (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1))) i)): 7 points increase in error, 5 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 n (/.f64 (*.f64 100 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1)) i))): 8 points increase in error, 5 points decrease in error
      (*.f64 n (Rewrite<= associate-*r/_binary64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)))): 5 points increase in error, 5 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 100 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i)) n)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 100 (*.f64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) i) n))): 7 points increase in error, 5 points decrease in error
      (*.f64 100 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)))): 37 points increase in error, 6 points decrease in error

   3. Taylor expanded in n around inf 58.3

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

   4. Taylor expanded in i around 0 9.3

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

   if 9.1999999999999997e-114 < i < 1e3

   1. Initial program 54.4

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

   2. Applied egg-rr 4.4

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

   3. Applied egg-rr 5.1

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

   4. Applied egg-rr 5.6

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot 0.01}{n}}} \]

   5. Applied egg-rr 7.0

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

   if 1e3 < i

   1. Initial program 30.4

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

   2. Applied egg-rr 41.9

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

   3. Taylor expanded in n around 0 22.1

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

   4. Simplified 22.1

      \[\leadsto \frac{\color{blue}{100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)}}{\frac{i}{n}} \]
      Proof
      (*.f64 100 (*.f64 n (-.f64 (log.f64 i) (log.f64 n)))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 i) (neg.f64 (log.f64 n)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (+.f64 (log.f64 i) (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 n)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (Rewrite<= +-commutative_binary64 (+.f64 (log.f64 (/.f64 1 n)) (log.f64 i))))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (+.f64 (Rewrite=> log-rec_binary64 (neg.f64 (log.f64 n))) (log.f64 i)))): 0 points increase in error, 0 points decrease in error
      (*.f64 100 (*.f64 n (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 n))) (log.f64 i)))): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot 0.01}{n}}\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{elif}\;i \leq 1000:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(\log i - \log n\right)\right)}{\frac{i}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.5
Cost	14032

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{if}\;i \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \end{array} \]

Alternative 2
Error	8.3
Cost	14032

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\\ \mathbf{if}\;i \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{t_0}{\frac{i \cdot 0.01}{n}}\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+35}:\\ \;\;\;\;t_0 \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \end{array} \]

Alternative 3
Error	13.1
Cost	7244

\[\begin{array}{l} t_0 := \frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{if}\;i \leq -1.06 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \end{array} \]

Alternative 4
Error	13.0
Cost	7244

\[\begin{array}{l} t_0 := n \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{if}\;i \leq -1.06 \cdot 10^{-123}:\\ \;\;\;\;\frac{t_0}{i \cdot 0.01}\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot t_0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \end{array} \]

Alternative 5
Error	13.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 0}{i}\\ \end{array} \]

Alternative 6
Error	20.8
Cost	1096

\[\begin{array}{l} t_0 := \frac{n \cdot 0}{i}\\ \mathbf{if}\;i \leq -0.058:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(\frac{-0.5}{n} + 0.5\right)\right)\right)\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	20.9
Cost	968

\[\begin{array}{l} t_0 := \frac{n \cdot 0}{i}\\ \mathbf{if}\;i \leq -7.1 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	23.7
Cost	844

\[\begin{array}{l} t_0 := \frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;i \cdot -50 + n \cdot 100\\ \mathbf{elif}\;i \leq 1.75 \cdot 10^{-35}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	20.9
Cost	844

\[\begin{array}{l} t_0 := \frac{n \cdot 0}{i}\\ \mathbf{if}\;i \leq -7.1 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;i \cdot -50 + n \cdot 100\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	20.9
Cost	844

\[\begin{array}{l} t_0 := \frac{n \cdot 0}{i}\\ \mathbf{if}\;i \leq -7.1 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;i \cdot -50 + n \cdot 100\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	28.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;i \leq 1.656792211520858 \cdot 10^{-238}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 12
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 13
Error	28.9
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2022300 
(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))))