Compound Interest

Average Error: 47.8 → 4.2

Time: 18.9s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{n \cdot t_0}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

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) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 0.0)
     (* n (/ (expm1 (* n (log1p (/ i n)))) (/ i 100.0)))
     (if (<= t_1 2e-19) (/ (* n t_0) (/ i 100.0)) (* n 100.0)))))

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) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) / (i / 100.0));
	} else if (t_1 <= 2e-19) {
		tmp = (n * t_0) / (i / 100.0);
	} else {
		tmp = n * 100.0;
	}
	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) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) / (i / 100.0));
	} else if (t_1 <= 2e-19) {
		tmp = (n * t_0) / (i / 100.0);
	} else {
		tmp = n * 100.0;
	}
	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) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) / (i / 100.0))
	elif t_1 <= 2e-19:
		tmp = (n * t_0) / (i / 100.0)
	else:
		tmp = n * 100.0
	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((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / 100.0)));
	elseif (t_1 <= 2e-19)
		tmp = Float64(Float64(n * t_0) / Float64(i / 100.0));
	else
		tmp = Float64(n * 100.0);
	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[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-19], N[(N[(n * t$95$0), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]

Target

Original	47.8
Target	47.3
Herbie	4.2

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.3

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

   2. Simplified 46.3

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

      [Start] 46.3	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-*r/ [=>] 46.3	\[ \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]
      sub-neg [=>] 46.3	\[ \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]
      metadata-eval [=>] 46.3	\[ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}\right)}{\frac{i}{n}} \]

   3. Applied egg-rr 46.3

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

   4. Applied egg-rr 39.3

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{i}\right)} - 1} \]

   5. Simplified 1.1

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

      [Start] 39.3	\[ e^{\mathsf{log1p}\left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{i}\right)} - 1 \]
      expm1-def [=>] 14.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{i}\right)\right)} \]
      expm1-log1p [=>] 1.1	\[ \color{blue}{n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{i}} \]
      associate-/l* [=>] 1.1	\[ n \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2e-19

   1. Initial program 3.7

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

   2. Simplified 3.7

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

      [Start] 3.7	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 3.7	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 3.7	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 3.7	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 3.7	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 3.7	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 3.7	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 3.7	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Applied egg-rr 3.5

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

   if 2e-19 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 61.0

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

   2. Simplified 60.0

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

      [Start] 61.0	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 60.0	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 60.0	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 60.0	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 60.0	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 60.0	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 60.0	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 60.0	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in i around 0 14.4

      \[\leadsto \color{blue}{100 \cdot n} \]

   4. Simplified 14.4

      \[\leadsto \color{blue}{n \cdot 100} \]
      Proof

      [Start] 14.4	\[ 100 \cdot n \]
      *-commutative [=>] 14.4	\[ \color{blue}{n \cdot 100} \]

3. Recombined 3 regimes into one program.

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternatives

Alternative 1
Error	11.4
Cost	13900

\[\begin{array}{l} \mathbf{if}\;i \leq 2.5 \cdot 10^{-208}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;i \leq 250:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 10^{+157}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot \left(\log i - \log n\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}{\frac{i}{100}}\\ \end{array} \]

Alternative 2
Error	12.5
Cost	7244

\[\begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.5:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) + -0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.5
Cost	7244

\[\begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-237}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 170:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) + -0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.7
Cost	1356

\[\begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-103}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.45:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) + -0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]

Alternative 5
Error	20.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;i \leq -1.12 \cdot 10^{+30} \lor \neg \left(i \leq 210\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]

Alternative 6
Error	20.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;i \leq -1.55 \lor \neg \left(i \leq 210\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 7
Error	22.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;i \leq -11500:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 220:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{\frac{1}{i}}\\ \end{array} \]

Alternative 8
Error	22.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;i \leq -11500:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 360:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{\frac{1}{i}}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -11500 \lor \neg \left(i \leq 215\right):\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 10
Error	27.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-248}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-237}:\\ \;\;\;\;16.666666666666668 \cdot \left(i \cdot \left(i \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 11
Error	27.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-249}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-238}:\\ \;\;\;\;16.666666666666668 \cdot \left(i \cdot \left(i \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 12
Error	27.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-250}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-240}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 13
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 14
Error	28.1
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2022364 
(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))))