Compound Interest

Average Error: 47.5 → 13.5

Time: 19.4s

Precision: binary64

Cost: 7816

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -1.26 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;\left(n \cdot \left(100 + 50 \cdot i\right) + -50 \cdot i\right) + 33.333333333333336 \cdot \frac{{i}^{2}}{n}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+212}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

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 -1.26e-26)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 8.6e-7)
     (+
      (+ (* n (+ 100.0 (* 50.0 i))) (* -50.0 i))
      (* 33.333333333333336 (/ (pow i 2.0) n)))
     (if (<= i 9.5e+212)
       (* 100.0 (/ n (/ i (+ (pow (+ 1.0 (/ i n)) n) -1.0))))
       0.0))))

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 tmp;
	if (i <= -1.26e-26) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 8.6e-7) {
		tmp = ((n * (100.0 + (50.0 * i))) + (-50.0 * i)) + (33.333333333333336 * (pow(i, 2.0) / n));
	} else if (i <= 9.5e+212) {
		tmp = 100.0 * (n / (i / (pow((1.0 + (i / n)), n) + -1.0)));
	} else {
		tmp = 0.0;
	}
	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 <= (-1.26d-26)) then
        tmp = 100.0d0 * ((exp(i) - 1.0d0) / (i / n))
    else if (i <= 8.6d-7) then
        tmp = ((n * (100.0d0 + (50.0d0 * i))) + ((-50.0d0) * i)) + (33.333333333333336d0 * ((i ** 2.0d0) / n))
    else if (i <= 9.5d+212) then
        tmp = 100.0d0 * (n / (i / (((1.0d0 + (i / n)) ** n) + (-1.0d0))))
    else
        tmp = 0.0d0
    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 <= -1.26e-26) {
		tmp = 100.0 * ((Math.exp(i) - 1.0) / (i / n));
	} else if (i <= 8.6e-7) {
		tmp = ((n * (100.0 + (50.0 * i))) + (-50.0 * i)) + (33.333333333333336 * (Math.pow(i, 2.0) / n));
	} else if (i <= 9.5e+212) {
		tmp = 100.0 * (n / (i / (Math.pow((1.0 + (i / n)), n) + -1.0)));
	} else {
		tmp = 0.0;
	}
	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):
	tmp = 0
	if i <= -1.26e-26:
		tmp = 100.0 * ((math.exp(i) - 1.0) / (i / n))
	elif i <= 8.6e-7:
		tmp = ((n * (100.0 + (50.0 * i))) + (-50.0 * i)) + (33.333333333333336 * (math.pow(i, 2.0) / n))
	elif i <= 9.5e+212:
		tmp = 100.0 * (n / (i / (math.pow((1.0 + (i / n)), n) + -1.0)))
	else:
		tmp = 0.0
	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)
	tmp = 0.0
	if (i <= -1.26e-26)
		tmp = Float64(100.0 * Float64(Float64(exp(i) - 1.0) / Float64(i / n)));
	elseif (i <= 8.6e-7)
		tmp = Float64(Float64(Float64(n * Float64(100.0 + Float64(50.0 * i))) + Float64(-50.0 * i)) + Float64(33.333333333333336 * Float64((i ^ 2.0) / n)));
	elseif (i <= 9.5e+212)
		tmp = Float64(100.0 * Float64(n / Float64(i / Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0))));
	else
		tmp = 0.0;
	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 <= -1.26e-26)
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	elseif (i <= 8.6e-7)
		tmp = ((n * (100.0 + (50.0 * i))) + (-50.0 * i)) + (33.333333333333336 * ((i ^ 2.0) / n));
	elseif (i <= 9.5e+212)
		tmp = 100.0 * (n / (i / (((1.0 + (i / n)) ^ n) + -1.0)));
	else
		tmp = 0.0;
	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, -1.26e-26], N[(100.0 * N[(N[(N[Exp[i], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.6e-7], N[(N[(N[(n * N[(100.0 + N[(50.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-50.0 * i), $MachinePrecision]), $MachinePrecision] + N[(33.333333333333336 * N[(N[Power[i, 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 9.5e+212], N[(100.0 * N[(n / N[(i / N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Target

Original	47.5
Target	47.4
Herbie	13.5

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

Derivation

1. Split input into 4 regimes

2. if i < -1.26000000000000002e-26

   1. Initial program 29.2

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

   2. Taylor expanded in n around inf 14.9

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

   if -1.26000000000000002e-26 < i < 8.6000000000000002e-7

   1. Initial program 58.4

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

   2. Taylor expanded in i around 0 14.1

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

   3. Simplified 14.1

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

      [Start] 14.1	\[ 100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) + \left(100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) \]
      rational.json-simplify-1 [=>] 14.1	\[ \color{blue}{\left(100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n\right) + 100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
      rational.json-simplify-43 [=>] 14.1	\[ \left(\color{blue}{n \cdot \left(\left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right)} + 100 \cdot n\right) + 100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]
      rational.json-simplify-51 [=>] 14.1	\[ \color{blue}{n \cdot \left(100 + \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right)} + 100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]
      rational.json-simplify-43 [=>] 14.1	\[ n \cdot \left(100 + \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right) + 100 \cdot \color{blue}{\left({i}^{2} \cdot \left(\left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) \cdot n\right)\right)} \]
      rational.json-simplify-43 [=>] 14.1	\[ n \cdot \left(100 + \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right) + \color{blue}{{i}^{2} \cdot \left(\left(\left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) \cdot n\right) \cdot 100\right)} \]
      rational.json-simplify-2 [=>] 14.1	\[ n \cdot \left(100 + \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right) + {i}^{2} \cdot \left(\color{blue}{\left(n \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)} \cdot 100\right) \]
      rational.json-simplify-1 [=>] 14.1	\[ n \cdot \left(100 + \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right) + {i}^{2} \cdot \left(\left(n \cdot \left(\color{blue}{\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right)} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right) \]
      rational.json-simplify-48 [=>] 14.1	\[ n \cdot \left(100 + \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right) + {i}^{2} \cdot \left(\left(n \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \cdot 100\right) \]

   4. Taylor expanded in n around 0 8.7

      \[\leadsto n \cdot \left(100 + \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right) + \color{blue}{33.333333333333336 \cdot \frac{{i}^{2}}{n}} \]

   5. Taylor expanded in n around 0 8.7

      \[\leadsto \color{blue}{\left(n \cdot \left(100 + 50 \cdot i\right) + -50 \cdot i\right)} + 33.333333333333336 \cdot \frac{{i}^{2}}{n} \]

   if 8.6000000000000002e-7 < i < 9.4999999999999993e212

   1. Initial program 32.7

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

   2. Simplified 32.7

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

      [Start] 32.7	\[ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
      rational.json-simplify-61 [=>] 32.7	\[ 100 \cdot \color{blue}{\frac{n}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]
      rational.json-simplify-7 [<=] 32.7	\[ 100 \cdot \frac{\color{blue}{\frac{n}{1}}}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}} \]
      rational.json-simplify-46 [<=] 32.7	\[ 100 \cdot \color{blue}{\frac{n}{1 \cdot \frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]
      rational.json-simplify-6 [=>] 32.7	\[ 100 \cdot \frac{n}{\color{blue}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]
      rational.json-simplify-16 [=>] 32.7	\[ 100 \cdot \frac{n}{\frac{i}{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

   if 9.4999999999999993e212 < i

   1. Initial program 31.8

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

   2. Taylor expanded in i around 0 29.4

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

   3. Taylor expanded in i around 0 29.4

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.26 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;\left(n \cdot \left(100 + 50 \cdot i\right) + -50 \cdot i\right) + 33.333333333333336 \cdot \frac{{i}^{2}}{n}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+212}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternatives

Alternative 1
Error	13.6
Cost	7692

\[\begin{array}{l} \mathbf{if}\;i \leq -1.26 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;\left(100 + \left(50 \cdot i + 16.666666666666668 \cdot {i}^{2}\right)\right) \cdot n\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+212}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7692

\[\begin{array}{l} \mathbf{if}\;i \leq -1.26 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot n + \left(16.666666666666668 \cdot {i}^{2} + i \cdot 50\right) \cdot n\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+212}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	13.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;i \leq -1.26 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;\left(100 + \left(50 \cdot i + 16.666666666666668 \cdot {i}^{2}\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	13.5
Cost	7108

\[\begin{array}{l} \mathbf{if}\;i \leq -1.26 \cdot 10^{-26}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;n \cdot 100 + n \cdot \left(i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	19.5
Cost	1352

\[\begin{array}{l} \mathbf{if}\;i \leq -3 \cdot 10^{-81}:\\ \;\;\;\;100 \cdot \left(\left(n - -1\right) + -1\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;n \cdot 100 + n \cdot \left(\left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	19.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;i \leq -2.15 \cdot 10^{-79}:\\ \;\;\;\;100 \cdot \left(\left(n - -1\right) + -1\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;n \cdot \left(100 + 50 \cdot i\right) + -50 \cdot i\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	19.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;i \leq -1.35 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \left(\left(n - -1\right) + -1\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;n \cdot 100 + n \cdot \left(i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	19.3
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \left(\left(n - -1\right) + -1\right)\\ \mathbf{if}\;i \leq -1.8 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 6.5 \cdot 10^{-49}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	19.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-83}:\\ \;\;\;\;100 \cdot \left(\left(n - -1\right) + -1\right)\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	20.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;i \leq -1.26 \cdot 10^{-26}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	62.1
Cost	192

\[-50 \cdot i \]

Alternative 12
Error	50.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023069 
(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))))