Compound Interest

Average Error: 47.6 → 0.5

Time: 17.7s

Precision: binary64

Cost: 21768

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq 100000000000:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot t_0\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

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) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 0.0)
     (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) (/ i n))
     (if (<= t_1 100000000000.0)
       (/ (* 100.0 (* n t_0)) i)
       (+ -1.0 (+ 1.0 (* 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 = (expm1((n * log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= 100000000000.0) {
		tmp = (100.0 * (n * t_0)) / i;
	} else {
		tmp = -1.0 + (1.0 + (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 = (Math.expm1((n * Math.log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= 100000000000.0) {
		tmp = (100.0 * (n * t_0)) / i;
	} else {
		tmp = -1.0 + (1.0 + (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 = (math.expm1((n * math.log1p((i / n)))) * 100.0) / (i / n)
	elif t_1 <= 100000000000.0:
		tmp = (100.0 * (n * t_0)) / i
	else:
		tmp = -1.0 + (1.0 + (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(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / Float64(i / n));
	elseif (t_1 <= 100000000000.0)
		tmp = Float64(Float64(100.0 * Float64(n * t_0)) / i);
	else
		tmp = Float64(-1.0 + Float64(1.0 + 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[(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, 100000000000.0], N[(N[(100.0 * N[(n * t$95$0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(-1.0 + N[(1.0 + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	47.6
Target	47.5
Herbie	0.5

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

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 46.1

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

   2. Applied egg-rr 0.3

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

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e11

   1. Initial program 3.1

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

   2. Simplified 3.1

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

      [Start] 3.1	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 3.1	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 3.1	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 3.1	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 3.1	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 3.1	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 3.1	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 3.1	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Applied egg-rr 3.2

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

   if 1e11 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 63.6

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

   2. Simplified 62.6

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

      [Start] 63.6	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      associate-/r/ [=>] 62.6	\[ 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]
      *-commutative [=>] 62.6	\[ 100 \cdot \color{blue}{\left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]
      *-rgt-identity [<=] 62.6	\[ 100 \cdot \left(\color{blue}{\left(n \cdot 1\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \]
      associate-*l* [=>] 62.6	\[ 100 \cdot \color{blue}{\left(n \cdot \left(1 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\right)} \]
      *-lft-identity [=>] 62.6	\[ 100 \cdot \left(n \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}\right) \]
      sub-neg [=>] 62.6	\[ 100 \cdot \left(n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i}\right) \]
      metadata-eval [=>] 62.6	\[ 100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i}\right) \]

   3. Taylor expanded in n around inf 62.6

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

   4. Simplified 15.6

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

      [Start] 62.6	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      associate-*r/ [=>] 62.6	\[ \color{blue}{\frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i}} \]
      associate-/l* [=>] 62.6	\[ \color{blue}{\frac{100}{\frac{i}{n \cdot \left(e^{i} - 1\right)}}} \]
      expm1-def [=>] 15.6	\[ \frac{100}{\frac{i}{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}} \]

   5. Taylor expanded in i around 0 14.9

      \[\leadsto \frac{100}{\color{blue}{\frac{1}{n}}} \]

   6. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\left(1 + 100 \cdot n\right) - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 100000000000:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 100000000000:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot t_0\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternative 2
Error	1.0
Cost	21768

\[\begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq 100000000000:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot t_0\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternative 3
Error	9.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+28} \lor \neg \left(n \leq 1.8 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \]

Alternative 4
Error	9.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+28} \lor \neg \left(n \leq 1.8 \cdot 10^{-52}\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \]

Alternative 5
Error	19.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;i \leq -8.5 \cdot 10^{-37} \lor \neg \left(i \leq 0.0018\right):\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 6
Error	19.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-165} \lor \neg \left(n \leq 5 \cdot 10^{-157}\right):\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \end{array} \]

Alternative 7
Error	15.2
Cost	832

\[\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)} \]

Alternative 8
Error	21.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -1.6 \lor \neg \left(i \leq 3500000000000\right):\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 9
Error	19.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-36} \lor \neg \left(i \leq 0.006\right):\\ \;\;\;\;-1 + \left(1 + n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 10
Error	21.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 3500000000000\right):\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 11
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 12
Error	28.4
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023057 
(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))))