Compound Interest

Percentage Accurate: 28.4% → 94.6%

Time: 15.9s

Alternatives: 14

Speedup: 9.5×

• could not determine a ground truth

  1. i = -2.0000000000000001e231
  2. n = 1.99999999999999997e77

Specification

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: 28.4% 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: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_0 INFINITY) (* t_0 100.0) (* n 100.0)))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -0.0

   1. Initial program 26.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 98.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 98.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   if -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.9%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 84.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 4.9 \cdot 10^{+115}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 4.9e+115)
   (* 100.0 (* n (/ (expm1 i) i)))
   (* (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) 100.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 4.9e+115) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else {
		tmp = ((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 4.9e+115) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else {
		tmp = ((Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 4.9e+115:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 4.9e+115)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 4.9e+115], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 4.89999999999999964e115

   1. Initial program 22.6%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 81.6%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 81.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 85.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 85.4%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   if 4.89999999999999964e115 < i

   1. Initial program 73.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4.9 \cdot 10^{+115}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -7.8 \cdot 10^{-246}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (/ (expm1 i) i)))))
   (if (<= n -7.8e-246) t_0 (if (<= n 1.5e-143) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (expm1(i) / i));
	double tmp;
	if (n <= -7.8e-246) {
		tmp = t_0;
	} else if (n <= 1.5e-143) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n * (Math.expm1(i) / i));
	double tmp;
	if (n <= -7.8e-246) {
		tmp = t_0;
	} else if (n <= 1.5e-143) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (math.expm1(i) / i))
	tmp = 0
	if n <= -7.8e-246:
		tmp = t_0
	elif n <= 1.5e-143:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -7.8e-246)
		tmp = t_0;
	elseif (n <= 1.5e-143)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7.8e-246], t$95$0, If[LessEqual[n, 1.5e-143], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7.79999999999999958e-246 or 1.49999999999999993e-143 < n

   1. Initial program 26.4%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 79.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 79.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 86.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   if -7.79999999999999958e-246 < n < 1.49999999999999993e-143

   1. Initial program 47.6%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. div-sub N/A

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

      15. clear-num N/A

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

   4. Applied egg-rr 23.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(i, -100\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 50.4%

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

   8. Taylor expanded in n around 0

      \[\leadsto \color{blue}{0} \]
   9. Step-by-step derivation

   10. Simplified 74.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.28 \cdot 10^{+29}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right) + \left(\frac{0.3333333333333333}{n \cdot n} + \left(\frac{-0.5}{n} + 0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.28e+29)
   (*
    100.0
    (+
     n
     (*
      i
      (+
       (* n (+ 0.5 (/ -0.5 n)))
       (*
        (+ (/ 0.3333333333333333 (* n n)) (+ (/ -0.5 n) 0.16666666666666666))
        (* i n))))))
   (if (<= n 1.6e-27)
     (* 100.0 (/ i (/ i n)))
     (/
      (*
       (* n 100.0)
       (*
        i
        (+
         1.0
         (*
          i
          (+ 0.5 (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
      i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.28e+29) {
		tmp = 100.0 * (n + (i * ((n * (0.5 + (-0.5 / n))) + (((0.3333333333333333 / (n * n)) + ((-0.5 / n) + 0.16666666666666666)) * (i * n)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.28d+29)) then
        tmp = 100.0d0 * (n + (i * ((n * (0.5d0 + ((-0.5d0) / n))) + (((0.3333333333333333d0 / (n * n)) + (((-0.5d0) / n) + 0.16666666666666666d0)) * (i * n)))))
    else if (n <= 1.6d-27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = ((n * 100.0d0) * (i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.28e+29) {
		tmp = 100.0 * (n + (i * ((n * (0.5 + (-0.5 / n))) + (((0.3333333333333333 / (n * n)) + ((-0.5 / n) + 0.16666666666666666)) * (i * n)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.28e+29:
		tmp = 100.0 * (n + (i * ((n * (0.5 + (-0.5 / n))) + (((0.3333333333333333 / (n * n)) + ((-0.5 / n) + 0.16666666666666666)) * (i * n)))))
	elif n <= 1.6e-27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.28e+29)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(Float64(n * Float64(0.5 + Float64(-0.5 / n))) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + Float64(Float64(-0.5 / n) + 0.16666666666666666)) * Float64(i * n))))));
	elseif (n <= 1.6e-27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(Float64(n * 100.0) * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.28e+29)
		tmp = 100.0 * (n + (i * ((n * (0.5 + (-0.5 / n))) + (((0.3333333333333333 / (n * n)) + ((-0.5 / n) + 0.16666666666666666)) * (i * n)))));
	elseif (n <= 1.6e-27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.28e+29], N[(100.0 * N[(n + N[(i * N[(N[(n * N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / n), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e-27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(n * 100.0), $MachinePrecision] * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.28e29

   1. Initial program 29.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) + \color{blue}{i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \left(\color{blue}{i} \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right), \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 56.0%

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

   if -1.28e29 < n < 1.59999999999999995e-27

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.0%

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

   if 1.59999999999999995e-27 < n

   1. Initial program 24.1%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 68.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 68.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 96.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(100 \cdot n\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(n \cdot 100\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 100\right), \left(e^{i} - 1\right)\right), i\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(100 \cdot n\right), \left(e^{i} - 1\right)\right), i\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(e^{i} - 1\right)\right), i\right) \]

      9. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\mathsf{expm1}\left(i\right)\right)\right), i\right) \]

      10. expm1-lowering-expm1.f64 96.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{expm1.f64}\left(i\right)\right), i\right) \]

   9. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right), i\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right), i\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right)\right), i\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right) \]

       8. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right) \]

   12. Simplified 79.7%

       \[\leadsto \frac{\left(100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}}{i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.28 \cdot 10^{+29}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right) + \left(\frac{0.3333333333333333}{n \cdot n} + \left(\frac{-0.5}{n} + 0.16666666666666666\right)\right) \cdot \left(i \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.95e+28)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n 1.6e-27)
     (* 100.0 (/ i (/ i n)))
     (/
      (*
       (* n 100.0)
       (*
        i
        (+
         1.0
         (*
          i
          (+ 0.5 (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
      i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.95e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.95d+28)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= 1.6d-27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = ((n * 100.0d0) * (i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.95e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.95e+28:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= 1.6e-27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.95e+28)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= 1.6e-27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(Float64(n * 100.0) * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.95e+28)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= 1.6e-27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = ((n * 100.0) * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.95e+28], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e-27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(n * 100.0), $MachinePrecision] * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.9500000000000001e28

   1. Initial program 29.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 71.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot i\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 56.0%

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

   if -2.9500000000000001e28 < n < 1.59999999999999995e-27

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.0%

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

   if 1.59999999999999995e-27 < n

   1. Initial program 24.1%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 68.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 68.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 96.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(100 \cdot n\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(n \cdot 100\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 100\right), \left(e^{i} - 1\right)\right), i\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(100 \cdot n\right), \left(e^{i} - 1\right)\right), i\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(e^{i} - 1\right)\right), i\right) \]

      9. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\mathsf{expm1}\left(i\right)\right)\right), i\right) \]

      10. expm1-lowering-expm1.f64 96.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{expm1.f64}\left(i\right)\right), i\right) \]

   9. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right), i\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right), i\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right)\right), i\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right) \]

       8. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right) \]

   12. Simplified 79.7%

       \[\leadsto \frac{\left(100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}}{i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.8:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.8e+28)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n 1.8)
     (* 100.0 (/ i (/ i n)))
     (*
      100.0
      (/
       (*
        n
        (*
         i
         (+
          1.0
          (*
           i
           (+ 0.5 (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
       i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.8e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-7.8d+28)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= 1.8d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((n * (i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.8e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.8e+28:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= 1.8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.8e+28)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= 1.8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -7.8e+28)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= 1.8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.8e+28], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.7999999999999997e28

   1. Initial program 29.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 71.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot i\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 56.0%

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

   if -7.7999999999999997e28 < n < 1.80000000000000004

   1. Initial program 29.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 69.3%

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

   if 1.80000000000000004 < n

   1. Initial program 26.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 95.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 95.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      8. *-lowering-*.f64 77.6%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 77.6%

      \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}}{i} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right) \cdot \left(i \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.9e+28)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n 1.45e-27)
     (* 100.0 (/ i (/ i n)))
     (*
      n
      (+
       100.0
       (*
        (+ 0.5 (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))
        (* i 100.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.9e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.45e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + ((0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))) * (i * 100.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.9d+28)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= 1.45d-27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + ((0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))) * (i * 100.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.9e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.45e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + ((0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))) * (i * 100.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.9e+28:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= 1.45e-27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + ((0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))) * (i * 100.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.9e+28)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= 1.45e-27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))) * Float64(i * 100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.9e+28)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= 1.45e-27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + ((0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))) * (i * 100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.9e+28], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.45e-27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.8999999999999999e28

   1. Initial program 29.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 71.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot i\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 56.0%

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

   if -1.8999999999999999e28 < n < 1.45000000000000002e-27

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.0%

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

   if 1.45000000000000002e-27 < n

   1. Initial program 24.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{n}^{2}} + i \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   5. Simplified 52.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + 100 \cdot \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + 100 \cdot \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot \color{blue}{\left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot \left(1 + \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \color{blue}{\left(1 + \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)\right)}\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + \color{blue}{\frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right), \color{blue}{\left(\frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right)\right)\right) \]

   8. Simplified 60.6%

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

   9. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot 1 + \color{blue}{100 \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(n, \left(100 + \color{blue}{100} \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(100 \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot \color{blue}{100}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(\left(\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) \cdot i\right) \cdot 100\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) \cdot \color{blue}{\left(i \cdot 100\right)}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) \cdot \left(100 \cdot \color{blue}{i}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right), \color{blue}{\left(100 \cdot i\right)}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right), \left(\color{blue}{100} \cdot i\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right), \left(100 \cdot i\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot i\right)\right)\right)\right), \left(100 \cdot i\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \frac{1}{24}\right)\right)\right)\right), \left(100 \cdot i\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right), \left(100 \cdot i\right)\right)\right)\right) \]

       14. *-lowering-*.f64 78.8%

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right), \mathsf{*.f64}\left(100, \color{blue}{i}\right)\right)\right)\right) \]

   11. Simplified 78.8%

       \[\leadsto n \cdot \color{blue}{\left(100 + \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right) \cdot \left(100 \cdot i\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right) \cdot \left(i \cdot 100\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.95e+28)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n 1.55e-27)
     (* 100.0 (/ i (/ i n)))
     (*
      n
      (*
       100.0
       (+
        1.0
        (*
         i
         (+
          0.5
          (* i (+ 0.16666666666666666 (* i 0.041666666666666664)))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.95e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.55e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.95d+28)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= 1.55d-27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.95e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.55e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.95e+28:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= 1.55e-27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.95e+28)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= 1.55e-27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.95e+28)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= 1.55e-27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.95e+28], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55e-27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.9500000000000001e28

   1. Initial program 29.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 71.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot i\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 56.0%

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

   if -2.9500000000000001e28 < n < 1.5499999999999999e-27

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.0%

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

   if 1.5499999999999999e-27 < n

   1. Initial program 24.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{n}^{2}} + i \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   5. Simplified 52.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + 100 \cdot \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + 100 \cdot \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot \color{blue}{\left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot \left(1 + \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \color{blue}{\left(1 + \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + \frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)\right)}\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) + \color{blue}{\frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right), \color{blue}{\left(\frac{i \cdot \left(i \cdot \left(\frac{-1}{4} \cdot i - \frac{1}{2}\right) - \frac{1}{2}\right)}{n}\right)}\right)\right)\right) \]

   8. Simplified 60.6%

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

   9. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 78.7%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 78.7%

       \[\leadsto n \cdot \left(100 \cdot \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.2e+28)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n 1.6e-27)
     (* 100.0 (/ i (/ i n)))
     (*
      100.0
      (*
       n
       (+
        1.0
        (*
         i
         (+
          0.5
          (* i (+ 0.16666666666666666 (* i 0.041666666666666664)))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.2e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-7.2d+28)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= 1.6d-27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.2e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.2e+28:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= 1.6e-27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.2e+28)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= 1.6e-27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -7.2e+28)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= 1.6e-27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.2e+28], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e-27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.1999999999999999e28

   1. Initial program 29.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 71.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot i\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 56.0%

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

   if -7.1999999999999999e28 < n < 1.59999999999999995e-27

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.0%

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

   if 1.59999999999999995e-27 < n

   1. Initial program 24.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{n}^{2}} + i \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   5. Simplified 52.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{24} \cdot i\right)}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 78.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 78.7%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 65.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.5e+28)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n 1.6e-27)
     (* 100.0 (/ i (/ i n)))
     (/ (* i (* n (+ 100.0 (* i 50.0)))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.5e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.5d+28)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= 1.6d-27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.5e+28) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.5e+28:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= 1.6e-27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.5e+28)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= 1.6e-27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -5.5e+28)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= 1.6e-27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.5e+28], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e-27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.5000000000000003e28

   1. Initial program 29.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 71.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 91.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \color{blue}{\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot i\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot i\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 56.0%

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

   if -5.5000000000000003e28 < n < 1.59999999999999995e-27

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.0%

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

   if 1.59999999999999995e-27 < n

   1. Initial program 24.1%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 68.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 68.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 96.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(100 \cdot n\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(n \cdot 100\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 100\right), \left(e^{i} - 1\right)\right), i\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(100 \cdot n\right), \left(e^{i} - 1\right)\right), i\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(e^{i} - 1\right)\right), i\right) \]

      9. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\mathsf{expm1}\left(i\right)\right)\right), i\right) \]

      10. expm1-lowering-expm1.f64 96.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{expm1.f64}\left(i\right)\right), i\right) \]

   9. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right)\right), i\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right)\right), i\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(i \cdot 50\right) \cdot n\right)\right), i\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

       8. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right)\right), i\right) \]

       12. *-lowering-*.f64 78.2%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right)\right), i\right) \]

   12. Simplified 78.2%

       \[\leadsto \frac{\color{blue}{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}}{i} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{if}\;n \leq -1.6 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* i (* n (+ 100.0 (* i 50.0)))) i)))
   (if (<= n -1.6e+28) t_0 (if (<= n 1.6e-27) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -1.6e+28) {
		tmp = t_0;
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_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 = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    if (n <= (-1.6d+28)) then
        tmp = t_0
    else if (n <= 1.6d-27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -1.6e+28) {
		tmp = t_0;
	} else if (n <= 1.6e-27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i
	tmp = 0
	if n <= -1.6e+28:
		tmp = t_0
	elif n <= 1.6e-27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i)
	tmp = 0.0
	if (n <= -1.6e+28)
		tmp = t_0;
	elseif (n <= 1.6e-27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	tmp = 0.0;
	if (n <= -1.6e+28)
		tmp = t_0;
	elseif (n <= 1.6e-27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.6e+28], t$95$0, If[LessEqual[n, 1.6e-27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.6e28 or 1.59999999999999995e-27 < n

   1. Initial program 26.3%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 70.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 70.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 94.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 94.0%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(100 \cdot n\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(n \cdot 100\right) \cdot \left(e^{i} - 1\right)\right), i\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 100\right), \left(e^{i} - 1\right)\right), i\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(100 \cdot n\right), \left(e^{i} - 1\right)\right), i\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(e^{i} - 1\right)\right), i\right) \]

      9. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\mathsf{expm1}\left(i\right)\right)\right), i\right) \]

      10. expm1-lowering-expm1.f64 93.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{expm1.f64}\left(i\right)\right), i\right) \]

   9. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{\frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right)\right), i\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right)\right), i\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(i \cdot 50\right) \cdot n\right)\right), i\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

       8. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right)\right), i\right) \]

       12. *-lowering-*.f64 67.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right)\right), i\right) \]

   12. Simplified 67.5%

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

   if -1.6e28 < n < 1.59999999999999995e-27

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 67.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 160000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.2e-11) 0.0 (if (<= i 160000.0) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.2e-11) {
		tmp = 0.0;
	} else if (i <= 160000.0) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.2d-11)) then
        tmp = 0.0d0
    else if (i <= 160000.0d0) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.2e-11) {
		tmp = 0.0;
	} else if (i <= 160000.0) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.2e-11:
		tmp = 0.0
	elif i <= 160000.0:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.2e-11)
		tmp = 0.0;
	elseif (i <= 160000.0)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.2e-11)
		tmp = 0.0;
	elseif (i <= 160000.0)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.2e-11], 0.0, If[LessEqual[i, 160000.0], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.2000000000000001e-11 or 1.6e5 < i

   1. Initial program 59.3%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. div-sub N/A

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

      15. clear-num N/A

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

   4. Applied egg-rr 55.1%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(i, -100\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 23.8%

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

   8. Taylor expanded in n around 0

      \[\leadsto \color{blue}{0} \]
   9. Step-by-step derivation

   10. Simplified 27.8%

       \[\leadsto \color{blue}{0} \]

   if -1.2000000000000001e-11 < i < 1.6e5

   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

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 89.0%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 160000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.2e-11) 0.0 (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.2e-11) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.2d-11)) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.2e-11) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.2e-11:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.2e-11)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.2e-11)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.2e-11], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.2000000000000001e-11

   1. Initial program 62.6%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      5. associate-+l- N/A

         \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

      6. neg-sub0 N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. div-sub N/A

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

      15. clear-num N/A

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

   4. Applied egg-rr 59.1%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(i, -100\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 27.3%

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

   8. Taylor expanded in n around 0

      \[\leadsto \color{blue}{0} \]
   9. Step-by-step derivation

   10. Simplified 30.5%

       \[\leadsto \color{blue}{0} \]

   if -1.2000000000000001e-11 < i

   1. Initial program 18.2%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 73.8%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 73.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(n \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(e^{i} - 1\right), \color{blue}{i}\right)\right)\right) \]

      4. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right)\right)\right) \]

      5. expm1-lowering-expm1.f64 81.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right)\right)\right) \]

   7. Simplified 81.4%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + 50 \cdot i\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot \color{blue}{50}\right)\right)\right) \]

      7. *-lowering-*.f64 73.8%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{50}\right)\right)\right) \]

   10. Simplified 73.8%

       \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 18.0% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 28.6%

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

   1. *-commutative N/A

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

   2. frac-2neg N/A

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

   3. associate-*l/ N/A

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

   4. neg-sub0 N/A

      \[\leadsto \frac{\left(0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

   5. associate-+l- N/A

      \[\leadsto \frac{\left(\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1\right) \cdot 100}{\mathsf{neg}\left(\frac{\color{blue}{i}}{n}\right)} \]

   6. neg-sub0 N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. frac-2neg N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. div-sub N/A

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

   15. clear-num N/A

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

4. Applied egg-rr 25.7%

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

5. Taylor expanded in i around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(i, -100\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 12.2%

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

8. Taylor expanded in n around 0

   \[\leadsto \color{blue}{0} \]
9. Step-by-step derivation

10. Simplified 15.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 34.8% 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 2024161 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))