Compound Interest

Average Error: 47.5 → 12.2

Time: 22.1s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -3 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 4.3 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot \left(n \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}}\\ \end{array} \]

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 -3e-6)
   (* 100.0 (/ (- (exp i) 1.0) (/ i n)))
   (if (<= i 4.3e-39)
     (* 100.0 (+ n (* (- 0.5 (* 0.5 (/ 1.0 n))) (* n i))))
     (* 100.0 (/ (* (- (log i) (log n)) n) (/ 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 <= -3e-6) {
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	} else if (i <= 4.3e-39) {
		tmp = 100.0 * (n + ((0.5 - (0.5 * (1.0 / n))) * (n * i)));
	} else {
		tmp = 100.0 * (((log(i) - log(n)) * n) / (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 <= (-3d-6)) then
        tmp = 100.0d0 * ((exp(i) - 1.0d0) / (i / n))
    else if (i <= 4.3d-39) then
        tmp = 100.0d0 * (n + ((0.5d0 - (0.5d0 * (1.0d0 / n))) * (n * i)))
    else
        tmp = 100.0d0 * (((log(i) - log(n)) * n) / (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 <= -3e-6) {
		tmp = 100.0 * ((Math.exp(i) - 1.0) / (i / n));
	} else if (i <= 4.3e-39) {
		tmp = 100.0 * (n + ((0.5 - (0.5 * (1.0 / n))) * (n * i)));
	} else {
		tmp = 100.0 * (((Math.log(i) - Math.log(n)) * n) / (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 <= -3e-6:
		tmp = 100.0 * ((math.exp(i) - 1.0) / (i / n))
	elif i <= 4.3e-39:
		tmp = 100.0 * (n + ((0.5 - (0.5 * (1.0 / n))) * (n * i)))
	else:
		tmp = 100.0 * (((math.log(i) - math.log(n)) * n) / (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 <= -3e-6)
		tmp = Float64(100.0 * Float64(Float64(exp(i) - 1.0) / Float64(i / n)));
	elseif (i <= 4.3e-39)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 * Float64(1.0 / n))) * Float64(n * i))));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(log(i) - log(n)) * n) / 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 <= -3e-6)
		tmp = 100.0 * ((exp(i) - 1.0) / (i / n));
	elseif (i <= 4.3e-39)
		tmp = 100.0 * (n + ((0.5 - (0.5 * (1.0 / n))) * (n * i)));
	else
		tmp = 100.0 * (((log(i) - log(n)) * n) / (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, -3e-6], N[(100.0 * N[(N[(N[Exp[i], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.3e-39], N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	47.5
Target	47.5
Herbie	12.2

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

Derivation

1. Split input into 3 regimes

2. if i < -3.0000000000000001e-6

   1. Initial program 27.4

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

   2. Taylor expanded in n around inf 12.0

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

   if -3.0000000000000001e-6 < i < 4.2999999999999999e-39

   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 8.4

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

   3. Simplified 8.4

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

      [Start] 8.4	\[ 100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]
      rational.json-simplify-43 [<=] 8.4	\[ 100 \cdot \left(n + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot \left(n \cdot i\right)}\right) \]

   if 4.2999999999999999e-39 < i

   1. Initial program 35.7

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

   2. Taylor expanded in n around 0 26.6

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

   3. Simplified 26.6

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

      [Start] 26.6	\[ 100 \cdot \frac{n \cdot \left(-1 \cdot \log n + \log i\right)}{\frac{i}{n}} \]
      rational.json-simplify-2 [=>] 26.6	\[ 100 \cdot \frac{n \cdot \left(\color{blue}{\log n \cdot -1} + \log i\right)}{\frac{i}{n}} \]
      rational.json-simplify-9 [=>] 26.6	\[ 100 \cdot \frac{n \cdot \left(\color{blue}{\left(-\log n\right)} + \log i\right)}{\frac{i}{n}} \]

   4. Taylor expanded in n around 0 26.6

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

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 4.3 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot \left(n \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.6
Cost	7108

\[\begin{array}{l} \mathbf{if}\;i \leq -4 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{e^{i} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot \left(n \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{i}{\frac{i}{n}} - 2\right) - -2\right)\\ \end{array} \]

Alternative 2
Error	22.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-26}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq -2.1 \cdot 10^{-236}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-147}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 3
Error	22.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-26}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq -1.2 \cdot 10^{-240}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-147}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 4
Error	23.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;n \leq -2.75 \cdot 10^{-236}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-147}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 5
Error	51.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023077 
(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))))