Compound Interest

Percentage Accurate: 25.6% → 98.5%

Time: 31.7s

Precision: binary64

Cost: 21896

• could not determine a ground truth

  1. i = 2.698693100047449e+196
  2. n = -3.095089555125207e+200

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 10^{-11}:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \end{array} \]

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)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) (/ i n))
     (if (<= t_1 1e-11)
       (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
       (/ 1.0 (* 0.01 (+ (/ 1.0 n) (* i (- (/ 0.5 (* n n)) (/ 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);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= 1e-11) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = 1.0 / (0.01 * ((1.0 / n) + (i * ((0.5 / (n * n)) - (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);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= 1e-11) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = 1.0 / (0.01 * ((1.0 / n) + (i * ((0.5 / (n * n)) - (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)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) * 100.0) / (i / n)
	elif t_1 <= 1e-11:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = 1.0 / (0.01 * ((1.0 / n) + (i * ((0.5 / (n * n)) - (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(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / Float64(i / n));
	elseif (t_1 <= 1e-11)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(1.0 / Float64(0.01 * Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 / Float64(n * n)) - Float64(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[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-11], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.01 * N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	25.6%
Target	26.9%
Herbie	98.5%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 25.1%

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

   2. Applied egg-rr 99.6%

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

      [Start] 25.1%	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 25.1%	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      *-commutative [=>] 25.1%	\[ \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]
      pow-to-exp [=>] 25.1%	\[ \frac{\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot 100}{\frac{i}{n}} \]
      expm1-def [=>] 40.4%	\[ \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot 100}{\frac{i}{n}} \]
      add-log-exp [=>] 25.1%	\[ \frac{\mathsf{expm1}\left(\color{blue}{\log \left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      pow-to-exp [<=] 25.1%	\[ \frac{\mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      log-pow [=>] 40.4%	\[ \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      log1p-udef [<=] 99.6%	\[ \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   if -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 9.99999999999999939e-12

   1. Initial program 93.5%

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

   2. Applied egg-rr 94.3%

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

      [Start] 93.5%	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      div-sub [=>] 94.3%	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]
      clear-num [<=] 94.3%	\[ 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]
      sub-neg [=>] 94.3%	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(-\frac{n}{i}\right)\right)} \]
      div-inv [=>] 94.3%	\[ 100 \cdot \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{1}{\frac{i}{n}}} + \left(-\frac{n}{i}\right)\right) \]
      clear-num [<=] 94.3%	\[ 100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \color{blue}{\frac{n}{i}} + \left(-\frac{n}{i}\right)\right) \]

   3. Simplified 94.3%

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

      [Start] 94.3%	\[ 100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} + \left(-\frac{n}{i}\right)\right) \]
      sub-neg [<=] 94.3%	\[ 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]

   if 9.99999999999999939e-12 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

   2. Applied egg-rr 1.9%

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

      [Start] 0.0%	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 0.0%	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 0.0%	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      distribute-lft-in [=>] 0.0%	\[ \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 0.0%	\[ \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]
      metadata-eval [=>] 0.0%	\[ \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]
      fma-udef [<=] 0.0%	\[ \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]
      associate-/r/ [=>] 1.9%	\[ \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i} \cdot n} \]
      *-commutative [<=] 1.9%	\[ \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
      clear-num [=>] 1.9%	\[ n \cdot \color{blue}{\frac{1}{\frac{i}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}} \]
      un-div-inv [=>] 1.9%	\[ \color{blue}{\frac{n}{\frac{i}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}} \]
      fma-udef [=>] 1.9%	\[ \frac{n}{\frac{i}{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}} \]
      metadata-eval [<=] 1.9%	\[ \frac{n}{\frac{i}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{100 \cdot -1}}} \]
      metadata-eval [<=] 1.9%	\[ \frac{n}{\frac{i}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{\left(-1\right)}}} \]
      distribute-lft-in [<=] 1.9%	\[ \frac{n}{\frac{i}{\color{blue}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}} \]
      sub-neg [<=] 1.9%	\[ \frac{n}{\frac{i}{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}} \]
      *-commutative [=>] 1.9%	\[ \frac{n}{\frac{i}{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}} \]

   3. Applied egg-rr 1.9%

      \[\leadsto \color{blue}{{\left(\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{n}\right)}^{-1}} \]
      Step-by-step derivation

      [Start] 1.9%	\[ \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}} \]
      clear-num [=>] 1.9%	\[ \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{n}}} \]
      inv-pow [=>] 1.9%	\[ \color{blue}{{\left(\frac{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{n}\right)}^{-1}} \]
      *-un-lft-identity [=>] 1.9%	\[ {\left(\frac{\frac{\color{blue}{1 \cdot i}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{n}\right)}^{-1} \]
      *-commutative [=>] 1.9%	\[ {\left(\frac{\frac{1 \cdot i}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}{n}\right)}^{-1} \]
      times-frac [=>] 1.9%	\[ {\left(\frac{\color{blue}{\frac{1}{100} \cdot \frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}{n}\right)}^{-1} \]
      metadata-eval [=>] 1.9%	\[ {\left(\frac{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{n}\right)}^{-1} \]

   4. Simplified 1.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}}} \]
      Step-by-step derivation

      [Start] 1.9%	\[ {\left(\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{n}\right)}^{-1} \]
      unpow-1 [=>] 1.9%	\[ \color{blue}{\frac{1}{\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{n}}} \]
      associate-/l* [=>] 1.9%	\[ \frac{1}{\color{blue}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}}}} \]

   5. Taylor expanded in i around 0 99.6%

      \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \frac{1}{n} + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)}} \]

   6. Simplified 99.6%

      \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}} \]
      Step-by-step derivation

      [Start] 99.6%	\[ \frac{1}{0.01 \cdot \frac{1}{n} + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} \]
      distribute-lft-out [=>] 99.6%	\[ \frac{1}{\color{blue}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)}} \]
      associate-*r/ [=>] 99.6%	\[ \frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right)} \]
      metadata-eval [=>] 99.6%	\[ \frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} \]
      unpow2 [=>] 99.6%	\[ \frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right)\right)} \]
      associate-*r/ [=>] 99.6%	\[ \frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)} \]
      metadata-eval [=>] 99.6%	\[ \frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-11}:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.5%
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 10^{-11}:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	96.2%
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_1 \leq 10^{-11}:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	97.6%
Cost	21896

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}\\ \mathbf{elif}\;t_1 \leq 10^{-11}:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	82.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+14} \lor \neg \left(n \leq 10^{-7}\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	82.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+14} \lor \neg \left(n \leq 0.000225\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	82.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+14}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 7
Accuracy	83.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;i \leq -45000000000000:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -6.5 \cdot 10^{-175} \lor \neg \left(i \leq 7 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 8
Accuracy	74.4%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;i \leq -1.6 \cdot 10^{-172} \lor \neg \left(i \leq 8.2 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{1}{0.01 \cdot \left(\frac{1}{n} + i \cdot \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 9
Accuracy	66.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;i \leq -3.6 \cdot 10^{-9} \lor \neg \left(i \leq 1.7 \cdot 10^{-34}\right):\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 10
Accuracy	64.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;i \leq -1.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.054:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{1}{\frac{i}{n}}\right)\\ \end{array} \]

Alternative 11
Accuracy	63.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.5 \cdot 10^{-15} \lor \neg \left(i \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 12
Accuracy	64.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.42 \lor \neg \left(i \leq 0.054\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13
Accuracy	66.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -0.1 \lor \neg \left(i \leq 1.7 \cdot 10^{-34}\right):\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14
Accuracy	63.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -1.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.054:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \end{array} \]

Alternative 15
Accuracy	3.0%
Cost	192

\[i \cdot -50 \]

Alternative 16
Accuracy	55.3%
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023263 
(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))))