Compound Interest

Average Error: 48.0 → 11.9

Time: 13.3s

Precision: binary64

Cost: 20236

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7.2:\\ \;\;\;\;i \cdot \left(50 \cdot n - 50\right) + 100 \cdot n\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+258}:\\ \;\;\;\;\frac{{n}^{2} \cdot \left(\log i - \log n\right)}{i} \cdot 100\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{+298}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

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 -2.9e-6)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 7.2)
     (+ (* i (- (* 50.0 n) 50.0)) (* 100.0 n))
     (if (<= i 2.9e+258)
       (* (/ (* (pow n 2.0) (- (log i) (log n))) i) 100.0)
       (if (<= i 1.4e+298)
         (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))
         (* 100.0 (/ i (/ 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 <= -2.9e-6) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 7.2) {
		tmp = (i * ((50.0 * n) - 50.0)) + (100.0 * n);
	} else if (i <= 2.9e+258) {
		tmp = ((pow(n, 2.0) * (log(i) - log(n))) / i) * 100.0;
	} else if (i <= 1.4e+298) {
		tmp = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * (i / (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 <= (-2.9d-6)) then
        tmp = 100.0d0 * ((exp(i) - 1.0d0) / (i / n))
    else if (i <= 7.2d0) then
        tmp = (i * ((50.0d0 * n) - 50.0d0)) + (100.0d0 * n)
    else if (i <= 2.9d+258) then
        tmp = (((n ** 2.0d0) * (log(i) - log(n))) / i) * 100.0d0
    else if (i <= 1.4d+298) then
        tmp = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
    else
        tmp = 100.0d0 * (i / (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 <= -2.9e-6) {
		tmp = 100.0 * ((Math.exp(i) - 1.0) / (i / n));
	} else if (i <= 7.2) {
		tmp = (i * ((50.0 * n) - 50.0)) + (100.0 * n);
	} else if (i <= 2.9e+258) {
		tmp = ((Math.pow(n, 2.0) * (Math.log(i) - Math.log(n))) / i) * 100.0;
	} else if (i <= 1.4e+298) {
		tmp = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * (i / (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 <= -2.9e-6:
		tmp = 100.0 * ((math.exp(i) - 1.0) / (i / n))
	elif i <= 7.2:
		tmp = (i * ((50.0 * n) - 50.0)) + (100.0 * n)
	elif i <= 2.9e+258:
		tmp = ((math.pow(n, 2.0) * (math.log(i) - math.log(n))) / i) * 100.0
	elif i <= 1.4e+298:
		tmp = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	else:
		tmp = 100.0 * (i / (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 <= -2.9e-6)
		tmp = Float64(100.0 * Float64(Float64(exp(i) - 1.0) / Float64(i / n)));
	elseif (i <= 7.2)
		tmp = Float64(Float64(i * Float64(Float64(50.0 * n) - 50.0)) + Float64(100.0 * n));
	elseif (i <= 2.9e+258)
		tmp = Float64(Float64(Float64((n ^ 2.0) * Float64(log(i) - log(n))) / i) * 100.0);
	elseif (i <= 1.4e+298)
		tmp = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(i / 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 <= -2.9e-6)
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	elseif (i <= 7.2)
		tmp = (i * ((50.0 * n) - 50.0)) + (100.0 * n);
	elseif (i <= 2.9e+258)
		tmp = (((n ^ 2.0) * (log(i) - log(n))) / i) * 100.0;
	elseif (i <= 1.4e+298)
		tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
	else
		tmp = 100.0 * (i / (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, -2.9e-6], N[(100.0 * N[(N[(N[Exp[i], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.2], N[(N[(i * N[(N[(50.0 * n), $MachinePrecision] - 50.0), $MachinePrecision]), $MachinePrecision] + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.9e+258], N[(N[(N[(N[Power[n, 2.0], $MachinePrecision] * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[i, 1.4e+298], 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[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	48.0
Target	47.8
Herbie	11.9

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

Derivation

1. Split input into 5 regimes

2. if i < -2.9000000000000002e-6

   1. Initial program 28.4

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

   2. Taylor expanded in n around inf 12.5

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

   if -2.9000000000000002e-6 < i < 7.20000000000000018

   1. Initial program 58.6

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

   2. Taylor expanded in i around 0 9.1

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

   3. Simplified 9.1

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

      [Start] 9.1	\[ 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n \]
      rational_best-simplify-1 [=>] 9.1	\[ \color{blue}{100 \cdot n + 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
      rational_best-simplify-3 [=>] 9.1	\[ \color{blue}{n \cdot 100} + 100 \cdot \left(n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]
      rational_best-simplify-113 [=>] 9.1	\[ n \cdot 100 + \color{blue}{n \cdot \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]

   4. Taylor expanded in n around 0 9.1

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

   5. Taylor expanded in i around 0 9.1

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

   if 7.20000000000000018 < i < 2.9000000000000001e258

   1. Initial program 30.8

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

   2. Taylor expanded in n around 0 20.4

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

   3. Simplified 20.4

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

      [Start] 20.4	\[ 100 \cdot \frac{{n}^{2} \cdot \left(-1 \cdot \log n + \log i\right)}{i} \]
      rational_best-simplify-3 [=>] 20.4	\[ \color{blue}{\frac{{n}^{2} \cdot \left(-1 \cdot \log n + \log i\right)}{i} \cdot 100} \]
      rational_best-simplify-3 [=>] 20.4	\[ \frac{{n}^{2} \cdot \left(\color{blue}{\log n \cdot -1} + \log i\right)}{i} \cdot 100 \]
      rational_best-simplify-17 [=>] 20.4	\[ \frac{{n}^{2} \cdot \left(\color{blue}{\left(-\log n\right)} + \log i\right)}{i} \cdot 100 \]

   4. Taylor expanded in n around 0 20.4

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

   if 2.9000000000000001e258 < i < 1.40000000000000008e298

   1. Initial program 35.2

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

   if 1.40000000000000008e298 < i

   1. Initial program 30.4

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

   2. Taylor expanded in i around 0 33.6

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7.2:\\ \;\;\;\;i \cdot \left(50 \cdot n - 50\right) + 100 \cdot n\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+258}:\\ \;\;\;\;\frac{{n}^{2} \cdot \left(\log i - \log n\right)}{i} \cdot 100\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{+298}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.9
Cost	13964

\[\begin{array}{l} \mathbf{if}\;i \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 0.0038:\\ \;\;\;\;i \cdot \left(50 \cdot n - 50\right) + 100 \cdot n\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+264}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\left(-\log n\right) + \log i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{+300}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 2
Error	14.0
Cost	7108

\[\begin{array}{l} t_0 := i \cdot \left(50 + \frac{1}{n} \cdot -50\right)\\ \mathbf{if}\;i \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \cdot \left(t_0 - 100\right) \ne 0:\\ \;\;\;\;\frac{n \cdot -10000}{50 \cdot i - 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(t_0 + 100\right)\\ \end{array} \]

Alternative 3
Error	19.8
Cost	1996

\[\begin{array}{l} t_0 := i \cdot \left(50 + \frac{1}{n} \cdot -50\right)\\ t_1 := \begin{array}{l} \mathbf{if}\;n \cdot \left(t_0 - 100\right) \ne 0:\\ \;\;\;\;\frac{n \cdot -10000}{50 \cdot i - 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(t_0 + 100\right)\\ \end{array}\\ \mathbf{if}\;n \leq -8.5 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-230}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	19.9
Cost	968

\[\begin{array}{l} t_0 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -0.88:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 0.52:\\ \;\;\;\;i \cdot \left(50 \cdot n - 50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	22.8
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 7.3 \cdot 10^{-50}:\\ \;\;\;\;100 \cdot \left(n + -0.5 \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.2
Cost	712

\[\begin{array}{l} t_0 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -12000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 9:\\ \;\;\;\;100 \cdot \left(n + -0.5 \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	62.1
Cost	192

\[i \cdot -50 \]

Alternative 8
Error	28.1
Cost	192

\[n \cdot 100 \]

Reproduce

herbie shell --seed 2023104 
(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))))