Compound Interest

Average Error: 47.9 → 12.5

Time: 17.1s

Precision: binary64

Cost: 14280

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00135:\\ \;\;\;\;\frac{e^{i} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;i \leq 3.2:\\ \;\;\;\;100 \cdot \left(\left(0.16666666666666666 \cdot {i}^{2} + \left(1 + \left(0.5 \cdot i + 0.041666666666666664 \cdot {i}^{3}\right)\right)\right) \cdot n\right)\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+245}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

↓

(FPCore (i n)
 :precision binary64
 (if (<= i -0.00135)
   (* (/ (- (* (exp i) n) n) i) 100.0)
   (if (<= i 3.2)
     (*
      100.0
      (*
       (+
        (* 0.16666666666666666 (pow i 2.0))
        (+ 1.0 (+ (* 0.5 i) (* 0.041666666666666664 (pow i 3.0)))))
       n))
     (if (<= i 5.8e+245)
       (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))
       (* 100.0 (/ 0.0 (/ 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 tmp;
	if (i <= -0.00135) {
		tmp = (((exp(i) * n) - n) / i) * 100.0;
	} else if (i <= 3.2) {
		tmp = 100.0 * (((0.16666666666666666 * pow(i, 2.0)) + (1.0 + ((0.5 * i) + (0.041666666666666664 * pow(i, 3.0))))) * n);
	} else if (i <= 5.8e+245) {
		tmp = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

↓

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-0.00135d0)) then
        tmp = (((exp(i) * n) - n) / i) * 100.0d0
    else if (i <= 3.2d0) then
        tmp = 100.0d0 * (((0.16666666666666666d0 * (i ** 2.0d0)) + (1.0d0 + ((0.5d0 * i) + (0.041666666666666664d0 * (i ** 3.0d0))))) * n)
    else if (i <= 5.8d+245) then
        tmp = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
    else
        tmp = 100.0d0 * (0.0d0 / (i / n))
    end if
    code = tmp
end function

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 tmp;
	if (i <= -0.00135) {
		tmp = (((Math.exp(i) * n) - n) / i) * 100.0;
	} else if (i <= 3.2) {
		tmp = 100.0 * (((0.16666666666666666 * Math.pow(i, 2.0)) + (1.0 + ((0.5 * i) + (0.041666666666666664 * Math.pow(i, 3.0))))) * n);
	} else if (i <= 5.8e+245) {
		tmp = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * (0.0 / (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):
	tmp = 0
	if i <= -0.00135:
		tmp = (((math.exp(i) * n) - n) / i) * 100.0
	elif i <= 3.2:
		tmp = 100.0 * (((0.16666666666666666 * math.pow(i, 2.0)) + (1.0 + ((0.5 * i) + (0.041666666666666664 * math.pow(i, 3.0))))) * n)
	elif i <= 5.8e+245:
		tmp = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	else:
		tmp = 100.0 * (0.0 / (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)
	tmp = 0.0
	if (i <= -0.00135)
		tmp = Float64(Float64(Float64(Float64(exp(i) * n) - n) / i) * 100.0);
	elseif (i <= 3.2)
		tmp = Float64(100.0 * Float64(Float64(Float64(0.16666666666666666 * (i ^ 2.0)) + Float64(1.0 + Float64(Float64(0.5 * i) + Float64(0.041666666666666664 * (i ^ 3.0))))) * n));
	elseif (i <= 5.8e+245)
		tmp = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

↓

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.00135)
		tmp = (((exp(i) * n) - n) / i) * 100.0;
	elseif (i <= 3.2)
		tmp = 100.0 * (((0.16666666666666666 * (i ^ 2.0)) + (1.0 + ((0.5 * i) + (0.041666666666666664 * (i ^ 3.0))))) * n);
	elseif (i <= 5.8e+245)
		tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
	else
		tmp = 100.0 * (0.0 / (i / n));
	end
	tmp_2 = 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_] := If[LessEqual[i, -0.00135], N[(N[(N[(N[(N[Exp[i], $MachinePrecision] * n), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[i, 3.2], N[(100.0 * N[(N[(N[(0.16666666666666666 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(0.5 * i), $MachinePrecision] + N[(0.041666666666666664 * N[Power[i, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.8e+245], 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], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	47.9
Target	47.5
Herbie	12.5

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

Derivation

1. Split input into 4 regimes

2. if i < -0.0013500000000000001

   1. Initial program 28.3

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

   2. Taylor expanded in n around inf 11.3

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

   3. Simplified 11.3

      \[\leadsto \color{blue}{\frac{e^{i} \cdot n - n}{i} \cdot 100} \]
      Proof

      [Start] 11.3	\[ 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.3	\[ \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 11.3	\[ \frac{\color{blue}{e^{i} \cdot n - n \cdot 1}}{i} \cdot 100 \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 11.3	\[ \frac{e^{i} \cdot n - \color{blue}{n}}{i} \cdot 100 \]

   if -0.0013500000000000001 < i < 3.2000000000000002

   1. Initial program 58.2

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

   2. Taylor expanded in i around 0 17.2

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) + \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n \cdot \left({i}^{3} \cdot \left(\left(0.041666666666666664 + 0.4583333333333333 \cdot \frac{1}{{n}^{2}}\right) - \left(0.25 \cdot \frac{1}{{n}^{3}} + 0.25 \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)} \]

   3. Simplified 17.2

      \[\leadsto 100 \cdot \color{blue}{\left(\left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n \cdot \left({i}^{3} \cdot \left(\frac{1}{{n}^{2}} \cdot 0.4583333333333333 + \left(0.041666666666666664 - \left(0.25 \cdot \frac{1}{{n}^{3}} + \frac{1}{n} \cdot 0.25\right)\right)\right)\right)\right) + \left(n + n \cdot \left({i}^{2} \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)\right)} \]
      Proof

      [Start] 17.2	\[ 100 \cdot \left(n + \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) + \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n \cdot \left({i}^{3} \cdot \left(\left(0.041666666666666664 + 0.4583333333333333 \cdot \frac{1}{{n}^{2}}\right) - \left(0.25 \cdot \frac{1}{{n}^{3}} + 0.25 \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 17.2	\[ 100 \cdot \left(n + \color{blue}{\left(\left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n \cdot \left({i}^{3} \cdot \left(\left(0.041666666666666664 + 0.4583333333333333 \cdot \frac{1}{{n}^{2}}\right) - \left(0.25 \cdot \frac{1}{{n}^{3}} + 0.25 \cdot \frac{1}{n}\right)\right)\right)\right) + n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 17.2	\[ 100 \cdot \color{blue}{\left(\left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + n \cdot \left({i}^{3} \cdot \left(\left(0.041666666666666664 + 0.4583333333333333 \cdot \frac{1}{{n}^{2}}\right) - \left(0.25 \cdot \frac{1}{{n}^{3}} + 0.25 \cdot \frac{1}{n}\right)\right)\right)\right) + \left(n + n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]

   4. Taylor expanded in n around inf 9.0

      \[\leadsto 100 \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot {i}^{2} + \left(1 + \left(0.5 \cdot i + 0.041666666666666664 \cdot {i}^{3}\right)\right)\right) \cdot n\right)} \]

   if 3.2000000000000002 < i < 5.8000000000000003e245

   1. Initial program 32.2

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

   if 5.8000000000000003e245 < i

   1. Initial program 32.0

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

   2. Taylor expanded in i around 0 30.1

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00135:\\ \;\;\;\;\frac{e^{i} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;i \leq 3.2:\\ \;\;\;\;100 \cdot \left(\left(0.16666666666666666 \cdot {i}^{2} + \left(1 + \left(0.5 \cdot i + 0.041666666666666664 \cdot {i}^{3}\right)\right)\right) \cdot n\right)\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+245}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	7692

\[\begin{array}{l} \mathbf{if}\;i \leq -0.000165:\\ \;\;\;\;\frac{e^{i} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;i \leq 2.5:\\ \;\;\;\;\left(i \cdot 50 + \left({i}^{2} \cdot 16.666666666666668 + 100\right)\right) \cdot n\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+245}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 2
Error	12.5
Cost	7692

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00016:\\ \;\;\;\;\frac{e^{i} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;i \leq 3.1:\\ \;\;\;\;\left(n + n \cdot \left(0.16666666666666666 \cdot {i}^{2} + i \cdot 0.5\right)\right) \cdot 100\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+245}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 3
Error	12.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;i \leq -0.00017:\\ \;\;\;\;\frac{e^{i} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;\left(i \cdot 50 + \left({i}^{2} \cdot 16.666666666666668 + 100\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 4
Error	12.5
Cost	7108

\[\begin{array}{l} \mathbf{if}\;i \leq -1.46 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.4:\\ \;\;\;\;n \cdot \left(100 + \left(i - i \cdot \frac{1}{n}\right) \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 5
Error	12.6
Cost	7108

\[\begin{array}{l} \mathbf{if}\;i \leq -1.46 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{i} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;i \leq 0.4:\\ \;\;\;\;n \cdot \left(100 + \left(i - i \cdot \frac{1}{n}\right) \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 6
Error	20.5
Cost	1096

\[\begin{array}{l} t_0 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.46 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.7:\\ \;\;\;\;n \cdot \left(100 + \left(i - i \cdot \frac{1}{n}\right) \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	20.6
Cost	840

\[\begin{array}{l} t_0 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.46 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(1 - -0.5 \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	22.6
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{-36}:\\ \;\;\;\;100 \cdot \left(n + -0.5 \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	22.3
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 3.2:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	20.6
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.46 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 12
Error	27.9
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023090 
(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))))