Compound Interest

Percentage Accurate: 28.3% → 31.5%

Time: 4.1s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. i = 6.053987814084295e-141
  2. n = 9.95263387496294e+37

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

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

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

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

Initial Program: 28.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

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

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

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

Alternative 1: 31.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ \mathbf{if}\;n \leq -3.3 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{\frac{i}{n}}}{\frac{i}{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+157}:\\ \;\;\;\;100 \cdot \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + t\_0\right)}^{n}}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)))
   (if (<= n -3.3e+226)
     (/ (/ (/ t_0 (/ i n)) (/ i n)) (/ i n))
     (if (<= n 1.65e+157)
       (* 100.0 (/ (- t_0 1.0) (/ i n)))
       (/ (pow (+ 1.0 t_0) n) (/ i n))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double tmp;
	if (n <= -3.3e+226) {
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	} else if (n <= 1.65e+157) {
		tmp = 100.0 * ((t_0 - 1.0) / (i / n));
	} else {
		tmp = pow((1.0 + t_0), n) / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (i / n)) ** n
    if (n <= (-3.3d+226)) then
        tmp = ((t_0 / (i / n)) / (i / n)) / (i / n)
    else if (n <= 1.65d+157) then
        tmp = 100.0d0 * ((t_0 - 1.0d0) / (i / n))
    else
        tmp = ((1.0d0 + t_0) ** n) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double tmp;
	if (n <= -3.3e+226) {
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	} else if (n <= 1.65e+157) {
		tmp = 100.0 * ((t_0 - 1.0) / (i / n));
	} else {
		tmp = Math.pow((1.0 + t_0), n) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	tmp = 0
	if n <= -3.3e+226:
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n)
	elif n <= 1.65e+157:
		tmp = 100.0 * ((t_0 - 1.0) / (i / n))
	else:
		tmp = math.pow((1.0 + t_0), n) / (i / n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	tmp = 0.0
	if (n <= -3.3e+226)
		tmp = Float64(Float64(Float64(t_0 / Float64(i / n)) / Float64(i / n)) / Float64(i / n));
	elseif (n <= 1.65e+157)
		tmp = Float64(100.0 * Float64(Float64(t_0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64((Float64(1.0 + t_0) ^ n) / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (1.0 + (i / n)) ^ n;
	tmp = 0.0;
	if (n <= -3.3e+226)
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	elseif (n <= 1.65e+157)
		tmp = 100.0 * ((t_0 - 1.0) / (i / n));
	else
		tmp = ((1.0 + t_0) ^ n) / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, If[LessEqual[n, -3.3e+226], N[(N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e+157], N[(100.0 * N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + t$95$0), $MachinePrecision], n], $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.29999999999999978e226

   1. Initial program 5.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in i around 0

      \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \]

   11. Applied rewrites 1.0%

       \[\leadsto \frac{{\left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}^{n}}{\frac{i}{n}} \]

   12. Taylor expanded in i around 0

       \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   14. Applied rewrites 24.8%

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

   if -3.29999999999999978e226 < n < 1.6500000000000001e157

   1. Initial program 31.7%

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

   if 1.6500000000000001e157 < n

   1. Initial program 18.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 16.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   8. Applied rewrites 17.8%

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

   9. Taylor expanded in i around 0

      \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \]

   11. Applied rewrites 36.6%

       \[\leadsto \frac{{\left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}^{n}}{\frac{i}{n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 31.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ \mathbf{if}\;n \leq -3.3 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{\frac{i}{n}}}{\frac{i}{n}}}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+172}:\\ \;\;\;\;100 \cdot \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + t\_0\right)}^{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)))
   (if (<= n -3.3e+226)
     (/ (/ (/ t_0 (/ i n)) (/ i n)) (/ i n))
     (if (<= n 1.65e+172)
       (* 100.0 (/ (- t_0 1.0) (/ i n)))
       (pow (+ 1.0 t_0) n)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double tmp;
	if (n <= -3.3e+226) {
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	} else if (n <= 1.65e+172) {
		tmp = 100.0 * ((t_0 - 1.0) / (i / n));
	} else {
		tmp = pow((1.0 + t_0), n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (i / n)) ** n
    if (n <= (-3.3d+226)) then
        tmp = ((t_0 / (i / n)) / (i / n)) / (i / n)
    else if (n <= 1.65d+172) then
        tmp = 100.0d0 * ((t_0 - 1.0d0) / (i / n))
    else
        tmp = (1.0d0 + t_0) ** n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double tmp;
	if (n <= -3.3e+226) {
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	} else if (n <= 1.65e+172) {
		tmp = 100.0 * ((t_0 - 1.0) / (i / n));
	} else {
		tmp = Math.pow((1.0 + t_0), n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	tmp = 0
	if n <= -3.3e+226:
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n)
	elif n <= 1.65e+172:
		tmp = 100.0 * ((t_0 - 1.0) / (i / n))
	else:
		tmp = math.pow((1.0 + t_0), n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	tmp = 0.0
	if (n <= -3.3e+226)
		tmp = Float64(Float64(Float64(t_0 / Float64(i / n)) / Float64(i / n)) / Float64(i / n));
	elseif (n <= 1.65e+172)
		tmp = Float64(100.0 * Float64(Float64(t_0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(1.0 + t_0) ^ n;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (1.0 + (i / n)) ^ n;
	tmp = 0.0;
	if (n <= -3.3e+226)
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	elseif (n <= 1.65e+172)
		tmp = 100.0 * ((t_0 - 1.0) / (i / n));
	else
		tmp = (1.0 + t_0) ^ n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, If[LessEqual[n, -3.3e+226], N[(N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e+172], N[(100.0 * N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 + t$95$0), $MachinePrecision], n], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.29999999999999978e226

   1. Initial program 5.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in i around 0

      \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \]

   11. Applied rewrites 1.0%

       \[\leadsto \frac{{\left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}^{n}}{\frac{i}{n}} \]

   12. Taylor expanded in i around 0

       \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   14. Applied rewrites 24.8%

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

   if -3.29999999999999978e226 < n < 1.64999999999999991e172

   1. Initial program 31.5%

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

   if 1.64999999999999991e172 < n

   1. Initial program 18.1%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 16.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   8. Applied rewrites 16.9%

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

   9. Taylor expanded in i around 0

      \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \]

   11. Applied rewrites 37.7%

       \[\leadsto \frac{{\left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}^{n}}{\frac{i}{n}} \]

   12. Taylor expanded in i around 0

       \[\leadsto n \]

   14. Applied rewrites 36.1%

       \[\leadsto {\left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}^{\color{blue}{n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 30.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := 100 \cdot \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \leq 7.8 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 6.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{\frac{i}{n}}}{\frac{i}{n}}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (* 100.0 (/ (- t_0 1.0) (/ i n)))))
   (if (<= i 7.8e-169)
     t_1
     (if (<= i 6.5e+195) (/ (/ (/ t_0 (/ i n)) (/ i n)) (/ i n)) t_1))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = 100.0 * ((t_0 - 1.0) / (i / n));
	double tmp;
	if (i <= 7.8e-169) {
		tmp = t_1;
	} else if (i <= 6.5e+195) {
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + (i / n)) ** n
    t_1 = 100.0d0 * ((t_0 - 1.0d0) / (i / n))
    if (i <= 7.8d-169) then
        tmp = t_1
    else if (i <= 6.5d+195) then
        tmp = ((t_0 / (i / n)) / (i / n)) / (i / n)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = 100.0 * ((t_0 - 1.0) / (i / n));
	double tmp;
	if (i <= 7.8e-169) {
		tmp = t_1;
	} else if (i <= 6.5e+195) {
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = 100.0 * ((t_0 - 1.0) / (i / n))
	tmp = 0
	if i <= 7.8e-169:
		tmp = t_1
	elif i <= 6.5e+195:
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n)
	else:
		tmp = t_1
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(100.0 * Float64(Float64(t_0 - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (i <= 7.8e-169)
		tmp = t_1;
	elseif (i <= 6.5e+195)
		tmp = Float64(Float64(Float64(t_0 / Float64(i / n)) / Float64(i / n)) / Float64(i / n));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (1.0 + (i / n)) ^ n;
	t_1 = 100.0 * ((t_0 - 1.0) / (i / n));
	tmp = 0.0;
	if (i <= 7.8e-169)
		tmp = t_1;
	elseif (i <= 6.5e+195)
		tmp = ((t_0 / (i / n)) / (i / n)) / (i / n);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 7.8e-169], t$95$1, If[LessEqual[i, 6.5e+195], N[(N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if i < 7.79999999999999954e-169 or 6.5000000000000003e195 < i

   1. Initial program 28.7%

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

   if 7.79999999999999954e-169 < i < 6.5000000000000003e195

   1. Initial program 22.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 19.7%

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

   6. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in i around 0

      \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \]

   11. Applied rewrites 23.3%

       \[\leadsto \frac{{\left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}^{n}}{\frac{i}{n}} \]

   12. Taylor expanded in i around 0

       \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   14. Applied rewrites 36.2%

       \[\leadsto \frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}}{\frac{i}{n}}}{\frac{\color{blue}{i}}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 28.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

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

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

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

Derivation

1. Initial program 27.1%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 17.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ \mathbf{if}\;i \leq -7.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{t\_0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)))
   (if (<= i -7.8e+170) (/ t_0 (/ i n)) (- t_0 1.0))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double tmp;
	if (i <= -7.8e+170) {
		tmp = t_0 / (i / n);
	} else {
		tmp = t_0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (i / n)) ** n
    if (i <= (-7.8d+170)) then
        tmp = t_0 / (i / n)
    else
        tmp = t_0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double tmp;
	if (i <= -7.8e+170) {
		tmp = t_0 / (i / n);
	} else {
		tmp = t_0 - 1.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	tmp = 0
	if i <= -7.8e+170:
		tmp = t_0 / (i / n)
	else:
		tmp = t_0 - 1.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	tmp = 0.0
	if (i <= -7.8e+170)
		tmp = Float64(t_0 / Float64(i / n));
	else
		tmp = Float64(t_0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (1.0 + (i / n)) ^ n;
	tmp = 0.0;
	if (i <= -7.8e+170)
		tmp = t_0 / (i / n);
	else
		tmp = t_0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, If[LessEqual[i, -7.8e+170], N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < -7.8000000000000005e170

   1. Initial program 85.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}{\frac{\color{blue}{i}}{n}} \]

   8. Applied rewrites 49.8%

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

   if -7.8000000000000005e170 < i

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

   5. Applied rewrites 14.0%

      \[\leadsto \color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 20.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

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

C

double code(double i, double n) {
	return (pow((1.0 + (i / n)), n) - 1.0) / (i / n);
}

Fortran

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

Java

public static double code(double i, double n) {
	return (Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n);
}

Python

def code(i, n):
	return (math.pow((1.0 + (i / n)), n) - 1.0) / (i / n)

Julia

function code(i, n)
	return Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n))
end

MATLAB

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

Wolfram

code[i_, n_] := N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.1%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 19.6%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]
6. Add Preprocessing

Alternative 7: 16.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ \mathbf{if}\;i \leq -6.5 \cdot 10^{+114}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))) (if (<= i -6.5e+114) t_0 (- t_0 1.0))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double tmp;
	if (i <= -6.5e+114) {
		tmp = t_0;
	} else {
		tmp = t_0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (i / n)) ** n
    if (i <= (-6.5d+114)) then
        tmp = t_0
    else
        tmp = t_0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double tmp;
	if (i <= -6.5e+114) {
		tmp = t_0;
	} else {
		tmp = t_0 - 1.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	tmp = 0
	if i <= -6.5e+114:
		tmp = t_0
	else:
		tmp = t_0 - 1.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	tmp = 0.0
	if (i <= -6.5e+114)
		tmp = t_0;
	else
		tmp = Float64(t_0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (1.0 + (i / n)) ^ n;
	tmp = 0.0;
	if (i <= -6.5e+114)
		tmp = t_0;
	else
		tmp = t_0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, If[LessEqual[i, -6.5e+114], t$95$0, N[(t$95$0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < -6.5000000000000001e114

   1. Initial program 66.3%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) + 100 \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)} \]

   5. Applied rewrites 26.7%

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

   if -6.5000000000000001e114 < i

   1. Initial program 22.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

   5. Applied rewrites 14.4%

      \[\leadsto \color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 11.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(1 + \frac{i}{n}\right)}^{n} \end{array} \]

FPCore

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

C

double code(double i, double n) {
	return pow((1.0 + (i / n)), n);
}

Fortran

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

Java

public static double code(double i, double n) {
	return Math.pow((1.0 + (i / n)), n);
}

Python

def code(i, n):
	return math.pow((1.0 + (i / n)), n)

Julia

function code(i, n)
	return Float64(1.0 + Float64(i / n)) ^ n
end

MATLAB

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

Wolfram

code[i_, n_] := N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]

Derivation

1. Initial program 27.1%

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

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) + 100 \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)} \]

5. Applied rewrites 10.9%

   \[\leadsto \color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} \]
6. Add Preprocessing

Alternative 9: 3.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{i}{n} \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (+ 1.0 (/ i n)))

C

double code(double i, double n) {
	return 1.0 + (i / n);
}

Fortran

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

Java

public static double code(double i, double n) {
	return 1.0 + (i / n);
}

Python

def code(i, n):
	return 1.0 + (i / n)

Julia

function code(i, n)
	return Float64(1.0 + Float64(i / n))
end

MATLAB

function tmp = code(i, n)
	tmp = 1.0 + (i / n);
end

Wolfram

code[i_, n_] := N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.1%

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

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

5. Applied rewrites 13.5%

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

6. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) + 100 \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)} \]

8. Applied rewrites 3.1%

   \[\leadsto \color{blue}{1 + \frac{i}{n}} \]
9. Add Preprocessing

Alternative 10: 2.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{i}{n} \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (/ i n))

C

double code(double i, double n) {
	return i / n;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i / n
end function

Java

public static double code(double i, double n) {
	return i / n;
}

Python

def code(i, n):
	return i / n

Julia

function code(i, n)
	return Float64(i / n)
end

MATLAB

function tmp = code(i, n)
	tmp = i / n;
end

Wolfram

code[i_, n_] := N[(i / n), $MachinePrecision]

Derivation

1. Initial program 27.1%

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

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

5. Applied rewrites 13.5%

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

6. Taylor expanded in i around 0

   \[\leadsto i \cdot \color{blue}{\left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{n}^{2}} + i \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)} \]

8. Applied rewrites 2.6%

   \[\leadsto \frac{i}{\color{blue}{n}} \]
9. Add Preprocessing

Developer Target 1: 33.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024321 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))