Compound Interest

Percentage Accurate: 27.6% → 99.2%

Time: 20.6s

Precision: binary64

Cost: 29388

• could not determine a ground truth

  1. i = -2.0000000000000001e231
  2. n = 1.99999999999999997e77

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0}{\frac{i}{100}} + n \cdot \frac{-1}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

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 -1e-18)
     (* 100.0 (+ (* (/ n i) 0.0) (- (* n (/ t_0 i)) (/ n i))))
     (if (<= t_1 0.0)
       (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) (/ i n))
       (if (<= t_1 INFINITY)
         (+ (* n (/ t_0 (/ i 100.0))) (* n (/ -1.0 (/ i 100.0))))
         (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

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 <= -1e-18) {
		tmp = 100.0 * (((n / i) * 0.0) + ((n * (t_0 / i)) - (n / i)));
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * (t_0 / (i / 100.0))) + (n * (-1.0 / (i / 100.0)));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	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 <= -1e-18) {
		tmp = 100.0 * (((n / i) * 0.0) + ((n * (t_0 / i)) - (n / i)));
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * Math.expm1((n * Math.log1p((i / n))))) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (n * (t_0 / (i / 100.0))) + (n * (-1.0 / (i / 100.0)));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	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 <= -1e-18:
		tmp = 100.0 * (((n / i) * 0.0) + ((n * (t_0 / i)) - (n / i)))
	elif t_1 <= 0.0:
		tmp = (100.0 * math.expm1((n * math.log1p((i / n))))) / (i / n)
	elif t_1 <= math.inf:
		tmp = (n * (t_0 / (i / 100.0))) + (n * (-1.0 / (i / 100.0)))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	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 <= -1e-18)
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) * 0.0) + Float64(Float64(n * Float64(t_0 / i)) - Float64(n / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * Float64(t_0 / Float64(i / 100.0))) + Float64(n * Float64(-1.0 / Float64(i / 100.0))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	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, -1e-18], N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] * 0.0), $MachinePrecision] + N[(N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(n * N[(t$95$0 / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(-1.0 / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	27.6%
Target	33.6%
Herbie	99.2%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -1.0000000000000001e-18

   1. Initial program 99.8%

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

   2. Applied egg-rr 100.0%

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

      [Start] 99.8%	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      div-sub [=>] 99.8%	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]
      associate-/r/ [=>] 99.8%	\[ 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} - \frac{1}{\frac{i}{n}}\right) \]
      clear-num [<=] 100.0%	\[ 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n - \color{blue}{\frac{n}{i}}\right) \]
      *-un-lft-identity [=>] 100.0%	\[ 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n - \color{blue}{1 \cdot \frac{n}{i}}\right) \]
      prod-diff [=>] 100.0%	\[ 100 \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right) + \mathsf{fma}\left(-\frac{n}{i}, 1, \frac{n}{i} \cdot 1\right)\right)} \]

   3. Simplified 100.0%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{n}{i} \cdot 0 + \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n - \frac{n}{i}\right)\right)} \]
      Step-by-step derivation

      [Start] 100.0%	\[ 100 \cdot \left(\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right) + \mathsf{fma}\left(-\frac{n}{i}, 1, \frac{n}{i} \cdot 1\right)\right) \]
      +-commutative [=>] 100.0%	\[ 100 \cdot \color{blue}{\left(\mathsf{fma}\left(-\frac{n}{i}, 1, \frac{n}{i} \cdot 1\right) + \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right)\right)} \]
      fma-udef [=>] 100.0%	\[ 100 \cdot \left(\color{blue}{\left(\left(-\frac{n}{i}\right) \cdot 1 + \frac{n}{i} \cdot 1\right)} + \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right)\right) \]
      distribute-lft-neg-in [<=] 100.0%	\[ 100 \cdot \left(\left(\color{blue}{\left(-\frac{n}{i} \cdot 1\right)} + \frac{n}{i} \cdot 1\right) + \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right)\right) \]
      distribute-rgt-neg-in [=>] 100.0%	\[ 100 \cdot \left(\left(\color{blue}{\frac{n}{i} \cdot \left(-1\right)} + \frac{n}{i} \cdot 1\right) + \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right)\right) \]
      metadata-eval [=>] 100.0%	\[ 100 \cdot \left(\left(\frac{n}{i} \cdot \color{blue}{-1} + \frac{n}{i} \cdot 1\right) + \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right)\right) \]
      distribute-lft-out [=>] 100.0%	\[ 100 \cdot \left(\color{blue}{\frac{n}{i} \cdot \left(-1 + 1\right)} + \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right)\right) \]
      metadata-eval [=>] 100.0%	\[ 100 \cdot \left(\frac{n}{i} \cdot \color{blue}{0} + \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, -\frac{n}{i} \cdot 1\right)\right) \]
      fma-udef [=>] 100.0%	\[ 100 \cdot \left(\frac{n}{i} \cdot 0 + \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n + \left(-\frac{n}{i} \cdot 1\right)\right)}\right) \]
      *-rgt-identity [=>] 100.0%	\[ 100 \cdot \left(\frac{n}{i} \cdot 0 + \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n + \left(-\color{blue}{\frac{n}{i}}\right)\right)\right) \]
      unsub-neg [=>] 100.0%	\[ 100 \cdot \left(\frac{n}{i} \cdot 0 + \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n - \frac{n}{i}\right)}\right) \]

   if -1.0000000000000001e-18 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 25.7%

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

   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] 25.7%	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 25.7%	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      *-commutative [=>] 25.7%	\[ \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]
      pow-to-exp [=>] 25.7%	\[ \frac{\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot 100}{\frac{i}{n}} \]
      expm1-def [=>] 37.4%	\[ \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot 100}{\frac{i}{n}} \]
      add-log-exp [=>] 25.7%	\[ \frac{\mathsf{expm1}\left(\color{blue}{\log \left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      pow-to-exp [<=] 25.7%	\[ \frac{\mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      log-pow [=>] 37.4%	\[ \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      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)) < +inf.0

   1. Initial program 98.2%

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

   2. Applied egg-rr 78.2%

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

      [Start] 98.2%	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 98.3%	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      *-commutative [=>] 98.3%	\[ \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]
      pow-to-exp [=>] 76.5%	\[ \frac{\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot 100}{\frac{i}{n}} \]
      expm1-def [=>] 78.2%	\[ \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot 100}{\frac{i}{n}} \]
      add-log-exp [=>] 76.5%	\[ \frac{\mathsf{expm1}\left(\color{blue}{\log \left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      pow-to-exp [<=] 98.3%	\[ \frac{\mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      log-pow [=>] 78.2%	\[ \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]
      log1p-udef [<=] 78.2%	\[ \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   3. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{\frac{i}{n}}{100}}} \]
      Step-by-step derivation

      [Start] 78.2%	\[ \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}} \]
      *-un-lft-identity [=>] 78.2%	\[ \color{blue}{1 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}} \]
      associate-/l* [=>] 78.1%	\[ 1 \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{\frac{i}{n}}{100}}} \]

   4. Simplified 78.1%

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

      [Start] 78.1%	\[ 1 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{\frac{i}{n}}{100}} \]
      *-lft-identity [=>] 78.1%	\[ \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{\frac{i}{n}}{100}}} \]
      associate-/l/ [=>] 78.1%	\[ \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{i}{100 \cdot n}}} \]

   5. Applied egg-rr 98.1%

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

      [Start] 78.1%	\[ \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100 \cdot n}} \]
      expm1-udef [=>] 76.5%	\[ \frac{\color{blue}{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1}}{\frac{i}{100 \cdot n}} \]
      div-sub [=>] 76.7%	\[ \color{blue}{\frac{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}{\frac{i}{100 \cdot n}} - \frac{1}{\frac{i}{100 \cdot n}}} \]
      *-commutative [=>] 76.7%	\[ \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}}}{\frac{i}{100 \cdot n}} - \frac{1}{\frac{i}{100 \cdot n}} \]
      log1p-udef [=>] 76.7%	\[ \frac{e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n}}{\frac{i}{100 \cdot n}} - \frac{1}{\frac{i}{100 \cdot n}} \]
      exp-to-pow [=>] 98.3%	\[ \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{\frac{i}{100 \cdot n}} - \frac{1}{\frac{i}{100 \cdot n}} \]
      associate-/r* [=>] 98.0%	\[ \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{\frac{i}{100}}{n}}} - \frac{1}{\frac{i}{100 \cdot n}} \]
      associate-/r* [=>] 98.1%	\[ \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{100}}{n}} - \frac{1}{\color{blue}{\frac{\frac{i}{100}}{n}}} \]

   6. Simplified 98.5%

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

      [Start] 98.1%	\[ \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{\frac{i}{100}}{n}} - \frac{1}{\frac{\frac{i}{100}}{n}} \]
      associate-/r/ [=>] 98.4%	\[ \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{100}} \cdot n} - \frac{1}{\frac{\frac{i}{100}}{n}} \]
      associate-/r/ [=>] 98.5%	\[ \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{100}} \cdot n - \color{blue}{\frac{1}{\frac{i}{100}} \cdot n} \]

   if +inf.0 < (/.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. Taylor expanded in n around inf 2.0%

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

   3. Simplified 84.6%

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

      [Start] 2.0%	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      *-commutative [=>] 2.0%	\[ \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]
      associate-/l* [=>] 2.0%	\[ \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]
      expm1-def [=>] 84.6%	\[ \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Taylor expanded in i around 0 100.0%

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

   5. Simplified 100.0%

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

      [Start] 100.0%	\[ \frac{n}{1 + -0.5 \cdot i} \cdot 100 \]
      *-commutative [=>] 100.0%	\[ \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

3. Recombined 4 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{100}} + n \cdot \frac{-1}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	29388

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0}{\frac{i}{100}} + n \cdot \frac{-1}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 2
Accuracy	98.3%
Cost	29388

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0}{\frac{i}{100}} + n \cdot \frac{-1}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	29388

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0}{\frac{i}{100}} + n \cdot \frac{-1}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 4
Accuracy	82.8%
Cost	7628

\[\begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -2.75 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-226}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(i \cdot i, 0.08333333333333333, i \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	75.8%
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{-202}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	79.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{-154} \lor \neg \left(n \leq 1.15 \cdot 10^{-177}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 7
Accuracy	80.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-154} \lor \neg \left(n \leq 8 \cdot 10^{-176}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 8
Accuracy	79.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-154} \lor \neg \left(n \leq 8 \cdot 10^{-181}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 9
Accuracy	65.2%
Cost	1484

\[\begin{array}{l} \mathbf{if}\;n \leq -2.35 \cdot 10^{+238}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -2.55 \cdot 10^{-153}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-177}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \]

Alternative 10
Accuracy	65.4%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+237}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.22 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-178}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\left(i \cdot i\right) \cdot 16.666666666666668 + \left(100 + i \cdot 50\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	59.3%
Cost	972

\[\begin{array}{l} \mathbf{if}\;n \leq -9.4 \cdot 10^{+47}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -5.6 \cdot 10^{-293}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-227}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 12
Accuracy	65.2%
Cost	972

\[\begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{+238}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.1 \cdot 10^{-151}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-177}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\\ \end{array} \]

Alternative 13
Accuracy	62.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-153} \lor \neg \left(n \leq 8.8 \cdot 10^{-181}\right):\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 14
Accuracy	63.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-154} \lor \neg \left(n \leq 4 \cdot 10^{-177}\right):\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 15
Accuracy	57.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+159}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	57.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -1.1 \cdot 10^{+158}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 17
Accuracy	58.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 18
Accuracy	54.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;i \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 19
Accuracy	2.8%
Cost	192

\[i \cdot -50 \]

Alternative 20
Accuracy	49.6%
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023178 
(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))))