Compound Interest

Percentage Accurate: 28.1% → 98.6%

Time: 29.8s

Alternatives: 14

Speedup: 38.0×

• 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: 28.1% 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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := 100 \cdot \frac{n \cdot t\_0 - n}{i}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-276}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-259}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (* 100.0 (/ (- (* n t_0) n) i))))
   (if (<= t_1 -2e-276)
     t_2
     (if (<= t_1 5e-259)
       (* 100.0 (* n (/ (expm1 (* n (log1p (/ i n)))) i)))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* 0.08333333333333333 (pow i 2.0)) (* i -0.5))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = 100.0 * (((n * t_0) - n) / i);
	double tmp;
	if (t_1 <= -2e-276) {
		tmp = t_2;
	} else if (t_1 <= 5e-259) {
		tmp = 100.0 * (n * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = 100.0 * (((n * t_0) - n) / i);
	double tmp;
	if (t_1 <= -2e-276) {
		tmp = t_2;
	} else if (t_1 <= 5e-259) {
		tmp = 100.0 * (n * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * Math.pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = 100.0 * (((n * t_0) - n) / i)
	tmp = 0
	if t_1 <= -2e-276:
		tmp = t_2
	elif t_1 <= 5e-259:
		tmp = 100.0 * (n * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * math.pow(i, 2.0)) + (i * -0.5))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(100.0 * Float64(Float64(Float64(n * t_0) - n) / i))
	tmp = 0.0
	if (t_1 <= -2e-276)
		tmp = t_2;
	elseif (t_1 <= 5e-259)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(0.08333333333333333 * (i ^ 2.0)) + Float64(i * -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-276], t$95$2, If[LessEqual[t$95$1, 5e-259], N[(100.0 * N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2e-276 or 4.99999999999999977e-259 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.8%

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

      1. associate-/r/ 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. metadata-eval 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/r/ 99.8%

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

      4. div-sub 99.6%

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

      5. clear-num 99.9%

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

      6. sub-neg 99.9%

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

      7. div-inv 99.7%

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

      8. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. sub-neg 99.9%

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

   8. Simplified 99.9%

      \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. sub-div 99.9%

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

      3. *-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -2e-276 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 4.99999999999999977e-259

   1. Initial program 19.8%

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

      1. associate-/r/ 18.7%

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

      2. sub-neg 18.7%

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

      3. metadata-eval 18.7%

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

   3. Simplified 18.7%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 18.7%

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

      2. metadata-eval 18.7%

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

      3. sub-neg 18.7%

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

      4. add-exp-log 18.7%

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

      5. expm1-def 18.7%

         \[\leadsto 100 \cdot \left(\frac{1 \cdot \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)}}{i} \cdot n\right) \]

      6. log-pow 32.1%

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

      7. log1p-udef 97.7%

         \[\leadsto 100 \cdot \left(\frac{1 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{i} \cdot n\right) \]

   6. Applied egg-rr 97.7%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{1 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{i} \cdot n\right) \]
   7. Step-by-step derivation

      1. *-lft-identity 97.7%

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

   8. Simplified 97.7%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. sub-neg 1.9%

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

      3. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. expm1-def 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-276}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-259}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-276}:\\ \;\;\;\;100 \cdot \frac{n \cdot t\_0 - n}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -2e-276)
     (* 100.0 (/ (- (* n t_0) n) i))
     (if (<= t_1 0.0)
       (* (/ (expm1 i) i) (* n 100.0))
       (if (<= t_1 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* 0.08333333333333333 (pow i 2.0)) (* i -0.5))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-276) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else if (t_1 <= 0.0) {
		tmp = (expm1(i) / i) * (n * 100.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-276) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else if (t_1 <= 0.0) {
		tmp = (Math.expm1(i) / i) * (n * 100.0);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * Math.pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -2e-276:
		tmp = 100.0 * (((n * t_0) - n) / i)
	elif t_1 <= 0.0:
		tmp = (math.expm1(i) / i) * (n * 100.0)
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * math.pow(i, 2.0)) + (i * -0.5))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-276)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * t_0) - n) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(i) / i) * Float64(n * 100.0));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(0.08333333333333333 * (i ^ 2.0)) + Float64(i * -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-276], N[(100.0 * N[(N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2e-276

   1. Initial program 99.8%

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

      1. associate-/r/ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. metadata-eval 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-/r/ 99.8%

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

      4. div-sub 99.4%

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

      5. clear-num 99.8%

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

      6. sub-neg 99.8%

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

      7. div-inv 99.4%

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

      8. clear-num 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

   8. Simplified 99.8%

      \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.8%

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

      2. sub-div 99.9%

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

      3. *-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -2e-276 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/r/ 18.4%

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

      3. associate-*l* 18.4%

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

      4. sub-neg 18.4%

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

      5. metadata-eval 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in n around inf 35.6%

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

      1. expm1-def 79.8%

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

   7. Simplified 79.8%

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

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

   1. Initial program 98.5%

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

      1. associate-/r/ 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

   3. Simplified 98.6%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. metadata-eval 98.6%

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

      2. sub-neg 98.6%

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

      3. associate-/r/ 98.5%

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

      4. div-sub 98.5%

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

      5. clear-num 98.6%

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

      6. sub-neg 98.6%

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

      7. div-inv 98.8%

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

      8. clear-num 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

   8. Simplified 98.8%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. sub-neg 1.9%

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

      3. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. expm1-def 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-276}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := 100 \cdot \frac{n \cdot t\_0 - n}{i}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-276}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (* 100.0 (/ (- (* n t_0) n) i))))
   (if (<= t_1 -2e-276)
     t_2
     (if (<= t_1 0.0)
       (* (/ (expm1 i) i) (* n 100.0))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* 0.08333333333333333 (pow i 2.0)) (* i -0.5))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = 100.0 * (((n * t_0) - n) / i);
	double tmp;
	if (t_1 <= -2e-276) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (expm1(i) / i) * (n * 100.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = 100.0 * (((n * t_0) - n) / i);
	double tmp;
	if (t_1 <= -2e-276) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (Math.expm1(i) / i) * (n * 100.0);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * Math.pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = 100.0 * (((n * t_0) - n) / i)
	tmp = 0
	if t_1 <= -2e-276:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = (math.expm1(i) / i) * (n * 100.0)
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * math.pow(i, 2.0)) + (i * -0.5))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(100.0 * Float64(Float64(Float64(n * t_0) - n) / i))
	tmp = 0.0
	if (t_1 <= -2e-276)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(i) / i) * Float64(n * 100.0));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(0.08333333333333333 * (i ^ 2.0)) + Float64(i * -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-276], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2e-276 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.1%

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

      1. associate-/r/ 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. metadata-eval 99.0%

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

      2. sub-neg 99.0%

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

      3. associate-/r/ 99.1%

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

      4. div-sub 98.9%

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

      5. clear-num 99.1%

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

      6. sub-neg 99.1%

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

      7. div-inv 99.1%

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

      8. clear-num 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. sub-neg 99.2%

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

   8. Simplified 99.2%

      \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.2%

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

      2. sub-div 99.3%

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

      3. *-commutative 99.3%

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

   10. Applied egg-rr 99.3%

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

   if -2e-276 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/r/ 18.4%

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

      3. associate-*l* 18.4%

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

      4. sub-neg 18.4%

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

      5. metadata-eval 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in n around inf 35.6%

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

      1. expm1-def 79.8%

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

   7. Simplified 79.8%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. sub-neg 1.9%

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

      3. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. expm1-def 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-276}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-23} \lor \neg \left(n \leq 2.1 \cdot 10^{-8}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.2e-23) (not (<= n 2.1e-8)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (* 100.0 (/ n (+ 1.0 (+ (* 0.08333333333333333 (pow i 2.0)) (* i -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.2e-23) || !(n <= 2.1e-8)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.2e-23) || !(n <= 2.1e-8)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * Math.pow(i, 2.0)) + (i * -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.2e-23) or not (n <= 2.1e-8):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 100.0 * (n / (1.0 + ((0.08333333333333333 * math.pow(i, 2.0)) + (i * -0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.2e-23) || !(n <= 2.1e-8))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(0.08333333333333333 * (i ^ 2.0)) + Float64(i * -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.2e-23], N[Not[LessEqual[n, 2.1e-8]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.19999999999999998e-23 or 2.09999999999999994e-8 < n

   1. Initial program 24.1%

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

      1. associate-/r/ 24.6%

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

      2. sub-neg 24.6%

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

      3. metadata-eval 24.6%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in n around inf 38.9%

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

      1. associate-/l* 38.9%

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

      2. expm1-def 91.9%

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

   7. Simplified 91.9%

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

   if -1.19999999999999998e-23 < n < 2.09999999999999994e-8

   1. Initial program 33.1%

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

      1. associate-/r/ 31.2%

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

      2. sub-neg 31.2%

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

      3. metadata-eval 31.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in n around inf 26.3%

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

      1. associate-/l* 26.3%

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

      2. expm1-def 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in i around 0 69.5%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-23} \lor \neg \left(n \leq 2.1 \cdot 10^{-8}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(0.08333333333333333 \cdot {i}^{2} + i \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-91} \lor \neg \left(n \leq 2.1 \cdot 10^{-8}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{i \cdot 2}{i + 2}}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.9e-91) (not (<= n 2.1e-8)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (* 100.0 (/ (/ (* i 2.0) (+ i 2.0)) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.9e-91) || !(n <= 2.1e-8)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 100.0 * (((i * 2.0) / (i + 2.0)) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.9e-91) || !(n <= 2.1e-8)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 100.0 * (((i * 2.0) / (i + 2.0)) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.9e-91) or not (n <= 2.1e-8):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 100.0 * (((i * 2.0) / (i + 2.0)) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.9e-91) || !(n <= 2.1e-8))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(i * 2.0) / Float64(i + 2.0)) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.9e-91], N[Not[LessEqual[n, 2.1e-8]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(i * 2.0), $MachinePrecision] / N[(i + 2.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9000000000000001e-91 or 2.09999999999999994e-8 < n

   1. Initial program 22.8%

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

      1. associate-/r/ 23.2%

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

      2. sub-neg 23.2%

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

      3. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in n around inf 36.6%

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

      1. associate-/l* 36.6%

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

      2. expm1-def 90.8%

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

   7. Simplified 90.8%

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

   if -2.9000000000000001e-91 < n < 2.09999999999999994e-8

   1. Initial program 36.7%

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

   3. Taylor expanded in i around 0 40.5%

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

      1. +-commutative 40.5%

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

   5. Simplified 40.5%

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

      1. flip-- 22.2%

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

      2. metadata-eval 22.2%

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

      3. difference-of-sqr-1 22.2%

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

      4. associate--l+ 47.2%

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

      5. metadata-eval 47.2%

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

      6. +-rgt-identity 47.2%

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

      7. associate-+l+ 47.2%

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

      8. metadata-eval 47.2%

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

      9. associate-+l+ 47.2%

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

      10. metadata-eval 47.2%

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

   7. Applied egg-rr 47.2%

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

   8. Taylor expanded in i around 0 65.9%

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{2 \cdot i}}{i + 2}}{\frac{i}{n}} \]
   9. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto 100 \cdot \frac{\frac{\color{blue}{i \cdot 2}}{i + 2}}{\frac{i}{n}} \]

   10. Simplified 65.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-91} \lor \neg \left(n \leq 2.1 \cdot 10^{-8}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{i \cdot 2}{i + 2}}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-287}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\frac{i \cdot 2}{i + 2}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.5e+200)
   (* 100.0 (/ (* i n) i))
   (if (<= n -5.2e-205)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 7e-287)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.1e-8)
         (* 100.0 (/ (/ (* i 2.0) (+ i 2.0)) (/ i n)))
         (* (* n 100.0) (+ 1.0 (* i 0.5))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.5e+200) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -5.2e-205) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7e-287) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.1e-8) {
		tmp = 100.0 * (((i * 2.0) / (i + 2.0)) / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.5d+200)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-5.2d-205)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 7d-287) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (n <= 2.1d-8) then
        tmp = 100.0d0 * (((i * 2.0d0) / (i + 2.0d0)) / (i / n))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.5e+200) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -5.2e-205) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7e-287) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.1e-8) {
		tmp = 100.0 * (((i * 2.0) / (i + 2.0)) / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.5e+200:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -5.2e-205:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 7e-287:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.1e-8:
		tmp = 100.0 * (((i * 2.0) / (i + 2.0)) / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.5e+200)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -5.2e-205)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 7e-287)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.1e-8)
		tmp = Float64(100.0 * Float64(Float64(Float64(i * 2.0) / Float64(i + 2.0)) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.5e+200)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -5.2e-205)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 7e-287)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (n <= 2.1e-8)
		tmp = 100.0 * (((i * 2.0) / (i + 2.0)) / (i / n));
	else
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.5e+200], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5.2e-205], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-287], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-8], N[(100.0 * N[(N[(N[(i * 2.0), $MachinePrecision] / N[(i + 2.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -9.49999999999999988e200

   1. Initial program 6.0%

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

   3. Taylor expanded in i around 0 9.8%

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

      1. +-commutative 9.8%

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

   5. Simplified 9.8%

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

      1. div-inv 9.8%

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

      2. clear-num 9.8%

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

      3. associate--l+ 30.4%

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

      4. metadata-eval 30.4%

         \[\leadsto 100 \cdot \left(\left(i + \color{blue}{0}\right) \cdot \frac{n}{i}\right) \]

      5. +-rgt-identity 30.4%

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

      6. *-commutative 30.4%

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

   7. Applied egg-rr 30.4%

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

      1. associate-*l/ 85.7%

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

   9. Applied egg-rr 85.7%

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

   if -9.49999999999999988e200 < n < -5.1999999999999997e-205

   1. Initial program 31.6%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in n around inf 24.1%

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

      1. associate-/l* 24.1%

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

      2. expm1-def 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in i around 0 67.7%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 67.7%

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

   10. Simplified 67.7%

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

   if -5.1999999999999997e-205 < n < 7e-287

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-/r/ 85.7%

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

      3. associate-*l* 85.7%

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

      4. sub-neg 85.7%

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

      5. metadata-eval 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in i around 0 85.7%

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

   if 7e-287 < n < 2.09999999999999994e-8

   1. Initial program 10.7%

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

   3. Taylor expanded in i around 0 24.2%

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

      1. +-commutative 24.2%

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

   5. Simplified 24.2%

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

      1. flip-- 8.5%

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

      2. metadata-eval 8.5%

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

      3. difference-of-sqr-1 8.5%

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

      4. associate--l+ 47.9%

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

      5. metadata-eval 47.9%

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

      6. +-rgt-identity 47.9%

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

      7. associate-+l+ 47.9%

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

      8. metadata-eval 47.9%

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

      9. associate-+l+ 47.9%

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

      10. metadata-eval 47.9%

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

   7. Applied egg-rr 47.9%

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

   8. Taylor expanded in i around 0 64.3%

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{2 \cdot i}}{i + 2}}{\frac{i}{n}} \]
   9. Step-by-step derivation

      1. *-commutative 64.3%

         \[\leadsto 100 \cdot \frac{\frac{\color{blue}{i \cdot 2}}{i + 2}}{\frac{i}{n}} \]

   10. Simplified 64.3%

       \[\leadsto 100 \cdot \frac{\frac{\color{blue}{i \cdot 2}}{i + 2}}{\frac{i}{n}} \]

   if 2.09999999999999994e-8 < n

   1. Initial program 22.7%

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

      1. *-commutative 22.7%

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

      2. associate-/r/ 23.2%

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

      3. associate-*l* 23.2%

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

      4. sub-neg 23.2%

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

      5. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in n around inf 42.7%

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

      1. expm1-def 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in i around 0 77.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot i\right)} \cdot \left(n \cdot 100\right) \]
   9. Step-by-step derivation

      1. *-commutative 77.7%

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

   10. Simplified 77.7%

       \[\leadsto \color{blue}{\left(1 + i \cdot 0.5\right)} \cdot \left(n \cdot 100\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-287}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\frac{i \cdot 2}{i + 2}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+199}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-204}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-287}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e+199)
   (* 100.0 (/ (* i n) i))
   (if (<= n -4.2e-204)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 7e-287)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.1e-8)
         (* 100.0 (/ i (/ i n)))
         (* (* n 100.0) (+ 1.0 (* i 0.5))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+199) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -4.2e-204) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7e-287) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.1e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.7d+199)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-4.2d-204)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 7d-287) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (n <= 2.1d-8) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+199) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -4.2e-204) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7e-287) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.1e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.7e+199:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -4.2e-204:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 7e-287:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.1e-8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e+199)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -4.2e-204)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 7e-287)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.1e-8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.7e+199)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -4.2e-204)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 7e-287)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (n <= 2.1e-8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e+199], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4.2e-204], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-287], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.70000000000000021e199

   1. Initial program 6.0%

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

   3. Taylor expanded in i around 0 9.8%

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

      1. +-commutative 9.8%

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

   5. Simplified 9.8%

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

      1. div-inv 9.8%

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

      2. clear-num 9.8%

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

      3. associate--l+ 30.4%

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

      4. metadata-eval 30.4%

         \[\leadsto 100 \cdot \left(\left(i + \color{blue}{0}\right) \cdot \frac{n}{i}\right) \]

      5. +-rgt-identity 30.4%

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

      6. *-commutative 30.4%

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

   7. Applied egg-rr 30.4%

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

      1. associate-*l/ 85.7%

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

   9. Applied egg-rr 85.7%

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

   if -3.70000000000000021e199 < n < -4.20000000000000018e-204

   1. Initial program 31.6%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in n around inf 24.1%

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

      1. associate-/l* 24.1%

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

      2. expm1-def 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in i around 0 67.7%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 67.7%

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

   10. Simplified 67.7%

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

   if -4.20000000000000018e-204 < n < 7e-287

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-/r/ 85.7%

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

      3. associate-*l* 85.7%

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

      4. sub-neg 85.7%

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

      5. metadata-eval 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in i around 0 85.7%

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

   if 7e-287 < n < 2.09999999999999994e-8

   1. Initial program 10.7%

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

   3. Taylor expanded in i around 0 24.2%

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

      1. +-commutative 24.2%

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

   5. Simplified 24.2%

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

   6. Taylor expanded in i around 0 63.6%

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

   if 2.09999999999999994e-8 < n

   1. Initial program 22.7%

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

      1. *-commutative 22.7%

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

      2. associate-/r/ 23.2%

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

      3. associate-*l* 23.2%

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

      4. sub-neg 23.2%

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

      5. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in n around inf 42.7%

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

      1. expm1-def 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in i around 0 77.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot i\right)} \cdot \left(n \cdot 100\right) \]
   9. Step-by-step derivation

      1. *-commutative 77.7%

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

   10. Simplified 77.7%

       \[\leadsto \color{blue}{\left(1 + i \cdot 0.5\right)} \cdot \left(n \cdot 100\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+199}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-204}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-287}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.5e+92)
   (* 100.0 (/ (* i n) i))
   (if (<= n 2.1e-8)
     (* 100.0 (/ i (/ i n)))
     (* (* n 100.0) (+ 1.0 (* i 0.5))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+92) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 2.1e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.5d+92)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 2.1d-8) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+92) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 2.1e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.5e+92:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 2.1e-8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.5e+92)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 2.1e-8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.5e+92)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 2.1e-8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.5e+92], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.50000000000000007e92

   1. Initial program 19.3%

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

   3. Taylor expanded in i around 0 6.3%

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

      1. +-commutative 6.3%

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

   5. Simplified 6.3%

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

      1. div-inv 6.3%

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

      2. clear-num 6.3%

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

      3. associate--l+ 30.6%

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

      4. metadata-eval 30.6%

         \[\leadsto 100 \cdot \left(\left(i + \color{blue}{0}\right) \cdot \frac{n}{i}\right) \]

      5. +-rgt-identity 30.6%

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

      6. *-commutative 30.6%

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

   7. Applied egg-rr 30.6%

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

      1. associate-*l/ 71.9%

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

   9. Applied egg-rr 71.9%

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

   if -1.50000000000000007e92 < n < 2.09999999999999994e-8

   1. Initial program 33.3%

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

   3. Taylor expanded in i around 0 28.7%

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

      1. +-commutative 28.7%

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

   5. Simplified 28.7%

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

   6. Taylor expanded in i around 0 65.8%

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

   if 2.09999999999999994e-8 < n

   1. Initial program 22.7%

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

      1. *-commutative 22.7%

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

      2. associate-/r/ 23.2%

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

      3. associate-*l* 23.2%

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

      4. sub-neg 23.2%

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

      5. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in n around inf 42.7%

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

      1. expm1-def 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in i around 0 77.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot i\right)} \cdot \left(n \cdot 100\right) \]
   9. Step-by-step derivation

      1. *-commutative 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{+93} \lor \neg \left(n \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.65e+93) (not (<= n 4e-8)))
   (* 100.0 (/ (* i n) i))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.65e+93) || !(n <= 4e-8)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.65d+93)) .or. (.not. (n <= 4d-8))) then
        tmp = 100.0d0 * ((i * n) / i)
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.65e+93) || !(n <= 4e-8)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.65e+93) or not (n <= 4e-8):
		tmp = 100.0 * ((i * n) / i)
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.65e+93) || !(n <= 4e-8))
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.65e+93) || ~((n <= 4e-8)))
		tmp = 100.0 * ((i * n) / i);
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.65e+93], N[Not[LessEqual[n, 4e-8]], $MachinePrecision]], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.65000000000000004e93 or 4.0000000000000001e-8 < n

   1. Initial program 21.8%

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

   3. Taylor expanded in i around 0 5.0%

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

      1. +-commutative 5.0%

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

   5. Simplified 5.0%

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

      1. div-inv 5.0%

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

      2. clear-num 5.0%

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

      3. associate--l+ 27.1%

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

      4. metadata-eval 27.1%

         \[\leadsto 100 \cdot \left(\left(i + \color{blue}{0}\right) \cdot \frac{n}{i}\right) \]

      5. +-rgt-identity 27.1%

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

      6. *-commutative 27.1%

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

   7. Applied egg-rr 27.1%

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

      1. associate-*l/ 73.8%

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

   9. Applied egg-rr 73.8%

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

   if -1.65000000000000004e93 < n < 4.0000000000000001e-8

   1. Initial program 33.1%

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

   3. Taylor expanded in i around 0 28.5%

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

      1. +-commutative 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in i around 0 66.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{+93} \lor \neg \left(n \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.5e+92)
   (* 100.0 (/ (* i n) i))
   (if (<= n 2.05e-8) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+92) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 2.05e-8) {
		tmp = 100.0 * (i / (i / n));
	} 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 (n <= (-1.5d+92)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 2.05d-8) then
        tmp = 100.0d0 * (i / (i / n))
    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 (n <= -1.5e+92) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 2.05e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.5e+92:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 2.05e-8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.5e+92)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 2.05e-8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	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 (n <= -1.5e+92)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 2.05e-8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.5e+92], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.05e-8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.50000000000000007e92

   1. Initial program 19.3%

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

   3. Taylor expanded in i around 0 6.3%

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

      1. +-commutative 6.3%

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

   5. Simplified 6.3%

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

      1. div-inv 6.3%

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

      2. clear-num 6.3%

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

      3. associate--l+ 30.6%

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

      4. metadata-eval 30.6%

         \[\leadsto 100 \cdot \left(\left(i + \color{blue}{0}\right) \cdot \frac{n}{i}\right) \]

      5. +-rgt-identity 30.6%

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

      6. *-commutative 30.6%

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

   7. Applied egg-rr 30.6%

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

      1. associate-*l/ 71.9%

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

   9. Applied egg-rr 71.9%

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

   if -1.50000000000000007e92 < n < 2.05000000000000016e-8

   1. Initial program 33.3%

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

   3. Taylor expanded in i around 0 28.7%

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

      1. +-commutative 28.7%

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

   5. Simplified 28.7%

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

   6. Taylor expanded in i around 0 65.8%

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

   if 2.05000000000000016e-8 < n

   1. Initial program 22.7%

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

      1. associate-/r/ 23.2%

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

      2. sub-neg 23.2%

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

      3. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in n around inf 42.7%

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

      1. associate-/l* 42.7%

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

      2. expm1-def 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in i around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. associate-*r* 77.7%

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

      3. distribute-rgt-out 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0)
   (* (/ n i) -200.0)
   (if (<= i 3.5e+40) (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n / i) * -200.0;
	} else if (i <= 3.5e+40) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 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 <= (-2.0d0)) then
        tmp = (n / i) * (-200.0d0)
    else if (i <= 3.5d+40) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n / i) * -200.0;
	} else if (i <= 3.5e+40) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = (n / i) * -200.0
	elif i <= 3.5e+40:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(Float64(n / i) * -200.0);
	elseif (i <= 3.5e+40)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = (n / i) * -200.0;
	elseif (i <= 3.5e+40)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision], If[LessEqual[i, 3.5e+40], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 72.2%

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

      1. associate-/r/ 68.2%

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

      2. sub-neg 68.2%

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

      3. metadata-eval 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in n around inf 74.2%

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

      1. associate-/l* 74.2%

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

      2. expm1-def 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in i around 0 39.8%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 39.8%

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

   10. Simplified 39.8%

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

   11. Taylor expanded in i around inf 39.8%

       \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]

   if -2 < i < 3.4999999999999999e40

   1. Initial program 7.1%

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

      1. associate-/r/ 7.6%

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

      2. sub-neg 7.6%

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

      3. metadata-eval 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in i around 0 81.8%

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

      1. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   if 3.4999999999999999e40 < i

   1. Initial program 44.3%

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

      1. *-commutative 44.3%

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

      2. associate-/r/ 44.6%

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

      3. associate-*l* 44.5%

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

      4. sub-neg 44.5%

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

      5. metadata-eval 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in n around inf 62.6%

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

      1. expm1-def 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in i around 0 51.5%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot i\right)} \cdot \left(n \cdot 100\right) \]
   9. Step-by-step derivation

      1. *-commutative 51.5%

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

   10. Simplified 51.5%

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

   11. Taylor expanded in i around inf 51.5%

       \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
   12. Step-by-step derivation

       1. *-commutative 51.5%

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

   13. Simplified 51.5%

       \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n \cdot -200}{i}\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0)
   (/ (* n -200.0) i)
   (if (<= i 3.5e+40) (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n * -200.0) / i;
	} else if (i <= 3.5e+40) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 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 <= (-2.0d0)) then
        tmp = (n * (-200.0d0)) / i
    else if (i <= 3.5d+40) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n * -200.0) / i;
	} else if (i <= 3.5e+40) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = (n * -200.0) / i
	elif i <= 3.5e+40:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(Float64(n * -200.0) / i);
	elseif (i <= 3.5e+40)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = (n * -200.0) / i;
	elseif (i <= 3.5e+40)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(N[(n * -200.0), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 3.5e+40], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 72.2%

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

      1. associate-/r/ 68.2%

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

      2. sub-neg 68.2%

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

      3. metadata-eval 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in n around inf 74.2%

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

      1. associate-/l* 74.2%

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

      2. expm1-def 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in i around 0 39.8%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 39.8%

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

   10. Simplified 39.8%

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

   11. Taylor expanded in i around inf 39.8%

       \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]
   12. Step-by-step derivation

       1. associate-*r/ 39.8%

          \[\leadsto \color{blue}{\frac{-200 \cdot n}{i}} \]

   13. Simplified 39.8%

       \[\leadsto \color{blue}{\frac{-200 \cdot n}{i}} \]

   if -2 < i < 3.4999999999999999e40

   1. Initial program 7.1%

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

      1. associate-/r/ 7.6%

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

      2. sub-neg 7.6%

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

      3. metadata-eval 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in i around 0 81.8%

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

      1. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   if 3.4999999999999999e40 < i

   1. Initial program 44.3%

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

      1. *-commutative 44.3%

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

      2. associate-/r/ 44.6%

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

      3. associate-*l* 44.5%

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

      4. sub-neg 44.5%

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

      5. metadata-eval 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in n around inf 62.6%

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

      1. expm1-def 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in i around 0 51.5%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot i\right)} \cdot \left(n \cdot 100\right) \]
   9. Step-by-step derivation

      1. *-commutative 51.5%

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

   10. Simplified 51.5%

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

   11. Taylor expanded in i around inf 51.5%

       \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
   12. Step-by-step derivation

       1. *-commutative 51.5%

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

   13. Simplified 51.5%

       \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n \cdot -200}{i}\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0) (* (/ n i) -200.0) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 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 (i <= (-2.0d0)) then
        tmp = (n / i) * (-200.0d0)
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = (n / i) * -200.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(Float64(n / i) * -200.0);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = (n / i) * -200.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2

   1. Initial program 72.2%

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

      1. associate-/r/ 68.2%

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

      2. sub-neg 68.2%

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

      3. metadata-eval 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in n around inf 74.2%

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

      1. associate-/l* 74.2%

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

      2. expm1-def 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in i around 0 39.8%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 39.8%

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

   10. Simplified 39.8%

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

   11. Taylor expanded in i around inf 39.8%

       \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]

   if -2 < i

   1. Initial program 16.2%

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

      1. associate-/r/ 16.7%

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

      2. sub-neg 16.7%

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

      3. metadata-eval 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in i around 0 63.6%

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

      1. *-commutative 63.6%

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

   7. Simplified 63.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.1% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * 100.0d0
end function

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 27.6%

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

   1. associate-/r/ 27.1%

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

   2. sub-neg 27.1%

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

   3. metadata-eval 27.1%

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

3. Simplified 27.1%

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

5. Taylor expanded in i around 0 51.8%

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

   1. *-commutative 51.8%

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

7. Simplified 51.8%

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

8. Final simplification 51.8%

   \[\leadsto n \cdot 100 \]
9. Add Preprocessing

Developer target: 34.2% 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 2024040 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

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