Compound Interest

Percentage Accurate: 28.4% → 98.9%

Time: 32.9s

Alternatives: 14

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 28.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-281}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{t\_0}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-263}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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-281)
     (* n (+ (* 100.0 (/ t_0 i)) (/ -100.0 i)))
     (if (<= t_1 1e-263)
       (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) (/ i n))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ (+ (pow (/ i n) n) -1.0) i)))
         (/
          1.0
          (+
           (* 0.01 (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))
           (* 0.01 (/ 1.0 n)))))))))

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-281) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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-281) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = (100.0 * Math.expm1((n * Math.log1p((i / n))))) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((Math.pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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-281:
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i))
	elif t_1 <= 1e-263:
		tmp = (100.0 * math.expm1((n * math.log1p((i / n))))) / (i / n)
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((math.pow((i / n), n) + -1.0) / i))
	else:
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)))
	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-281)
		tmp = Float64(n * Float64(Float64(100.0 * Float64(t_0 / i)) + Float64(-100.0 / i)));
	elseif (t_1 <= 1e-263)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(i / n) ^ n) + -1.0) / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))) + Float64(0.01 * Float64(1.0 / n))));
	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-281], N[(n * N[(N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] + N[(-100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-263], N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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-281

   1. Initial program 99.4%

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

      1. associate-*r/ 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.6%

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

      4. metadata-eval 99.6%

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

      5. sub-neg 99.6%

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

      6. *-commutative 99.6%

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

      7. add-exp-log 99.6%

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

      8. expm1-define 99.6%

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

      9. log-pow 63.2%

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

      10. log1p-define 63.2%

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

   6. Applied egg-rr 63.2%

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

      1. clear-num 63.4%

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

      2. inv-pow 63.4%

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

      3. *-commutative 63.4%

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

      4. associate-/r* 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. unpow-1 63.4%

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

      2. associate-/l/ 63.4%

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

      3. *-commutative 63.4%

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

   10. Simplified 63.4%

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

       1. associate-/r* 63.4%

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

       2. associate-/r/ 63.4%

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

       3. associate-/l/ 63.1%

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

       4. *-commutative 63.1%

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

   12. Applied egg-rr 63.1%

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

       1. associate-*l/ 63.1%

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

       2. *-lft-identity 63.1%

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

   14. Simplified 63.1%

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

       1. associate-/r/ 63.2%

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

       2. expm1-undefine 63.2%

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

       3. *-commutative 63.2%

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

       4. log1p-undefine 63.2%

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

       5. pow-to-exp 99.6%

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

       6. sub-div 99.6%

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

       7. *-commutative 99.6%

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

       8. sub-neg 99.6%

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

       9. distribute-lft-in 99.4%

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

       10. +-commutative 99.4%

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

       11. distribute-neg-frac 99.4%

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

       12. metadata-eval 99.4%

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

   16. Applied egg-rr 99.4%

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

       1. associate-*l* 99.4%

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

       2. associate-*l* 99.6%

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

       3. distribute-lft-out 99.8%

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

       4. +-commutative 99.8%

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

       5. associate-*r/ 100.0%

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

       6. metadata-eval 100.0%

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

   18. Simplified 100.0%

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

   if -2e-281 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-263

   1. Initial program 22.5%

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

      1. associate-*r/ 22.5%

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

      2. sub-neg 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. metadata-eval 22.5%

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

      5. metadata-eval 22.5%

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

      6. metadata-eval 22.5%

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

      7. fma-define 22.5%

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

      8. metadata-eval 22.5%

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

   3. Simplified 22.5%

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

      1. fma-undefine 22.5%

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

      2. metadata-eval 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. metadata-eval 22.5%

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

      5. sub-neg 22.5%

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

      6. *-commutative 22.5%

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

      7. add-exp-log 22.5%

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

      8. expm1-define 22.5%

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

      9. log-pow 34.1%

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

      10. log1p-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 1e-263 < (/.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%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. sub-neg 99.7%

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

      6. *-commutative 99.7%

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

      7. add-exp-log 99.7%

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

      8. expm1-define 99.7%

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

      9. log-pow 49.9%

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

      10. log1p-define 49.9%

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

   6. Applied egg-rr 49.9%

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

      1. clear-num 50.0%

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

      2. inv-pow 50.0%

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

      3. *-commutative 50.0%

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

      4. associate-/r* 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow-1 50.0%

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

      2. associate-/l/ 50.0%

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

      3. *-commutative 50.0%

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

   10. Simplified 50.0%

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

   11. Taylor expanded in i around inf 37.5%

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

       1. associate-/l* 37.5%

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

   13. Simplified 99.8%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-define 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

      1. fma-undefine 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. sub-neg 0.0%

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

      6. *-commutative 0.0%

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

      7. add-exp-log 0.0%

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

      8. expm1-define 0.0%

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

      9. log-pow 0.0%

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

      10. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/r* 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. unpow-1 0.0%

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

      2. associate-/l/ 0.0%

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

      3. *-commutative 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in i around 0 99.5%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-263}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := \frac{t\_0}{i \cdot \frac{0.01}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-315}:\\ \;\;\;\;t\_2 - n \cdot \frac{100}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2 - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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 (/ t_0 (* i (/ 0.01 n)))))
   (if (<= t_1 -2e-315)
     (- t_2 (* n (/ 100.0 i)))
     (if (<= t_1 0.0)
       (* 100.0 (* n (/ (expm1 i) i)))
       (if (<= t_1 INFINITY)
         (- t_2 (/ (* n 100.0) i))
         (/
          1.0
          (+
           (* 0.01 (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))
           (* 0.01 (/ 1.0 n)))))))))

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 = t_0 / (i * (0.01 / n));
	double tmp;
	if (t_1 <= -2e-315) {
		tmp = t_2 - (n * (100.0 / i));
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2 - ((n * 100.0) / i);
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 = t_0 / (i * (0.01 / n));
	double tmp;
	if (t_1 <= -2e-315) {
		tmp = t_2 - (n * (100.0 / i));
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2 - ((n * 100.0) / i);
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 = t_0 / (i * (0.01 / n))
	tmp = 0
	if t_1 <= -2e-315:
		tmp = t_2 - (n * (100.0 / i))
	elif t_1 <= 0.0:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	elif t_1 <= math.inf:
		tmp = t_2 - ((n * 100.0) / i)
	else:
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)))
	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(t_0 / Float64(i * Float64(0.01 / n)))
	tmp = 0.0
	if (t_1 <= -2e-315)
		tmp = Float64(t_2 - Float64(n * Float64(100.0 / i)));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(t_2 - Float64(Float64(n * 100.0) / i));
	else
		tmp = Float64(1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))) + Float64(0.01 * Float64(1.0 / n))));
	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[(t$95$0 / N[(i * N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-315], N[(t$95$2 - N[(n * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$2 - N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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)) < -2.0000000019e-315

   1. Initial program 95.0%

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

      1. associate-*r/ 95.2%

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

      2. sub-neg 95.2%

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

      3. distribute-lft-in 95.2%

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

      4. metadata-eval 95.2%

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

      5. metadata-eval 95.2%

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

      6. metadata-eval 95.2%

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

      7. fma-define 95.3%

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

      8. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

      1. fma-undefine 95.2%

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

      2. metadata-eval 95.2%

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

      3. distribute-lft-in 95.2%

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

      4. metadata-eval 95.2%

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

      5. sub-neg 95.2%

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

      6. *-commutative 95.2%

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

      7. add-exp-log 95.2%

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

      8. expm1-define 95.2%

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

      9. log-pow 66.3%

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

      10. log1p-define 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. clear-num 61.8%

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

      2. inv-pow 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-/r* 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. unpow-1 61.8%

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

      2. associate-/l/ 61.8%

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

      3. *-commutative 61.8%

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

   10. Simplified 61.8%

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

       1. associate-/r* 61.8%

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

       2. associate-/r/ 66.4%

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

       3. associate-/l/ 66.1%

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

       4. *-commutative 66.1%

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

   12. Applied egg-rr 66.1%

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

       1. associate-*l/ 66.1%

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

       2. *-lft-identity 66.1%

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

   14. Simplified 66.1%

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

       1. expm1-undefine 61.7%

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

       2. *-commutative 61.7%

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

       3. log1p-undefine 61.7%

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

       4. pow-to-exp 95.0%

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

       5. div-sub 95.0%

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

       6. +-commutative 95.0%

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

       7. div-inv 95.0%

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

       8. *-commutative 95.0%

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

       9. associate-/r* 95.0%

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

       10. metadata-eval 95.0%

           \[\leadsto \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i \cdot \frac{\color{blue}{0.01}}{n}} - \frac{1}{\frac{i}{n \cdot 100}} \]

       11. clear-num 95.1%

           \[\leadsto \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i \cdot \frac{0.01}{n}} - \color{blue}{\frac{n \cdot 100}{i}} \]

   16. Applied egg-rr 95.1%

       \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i \cdot \frac{0.01}{n}} - \frac{n \cdot 100}{i}} \]
   17. Step-by-step derivation

       1. +-commutative 95.1%

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i \cdot \frac{0.01}{n}} - \frac{n \cdot 100}{i} \]

       2. associate-/l* 95.4%

          \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i \cdot \frac{0.01}{n}} - \color{blue}{n \cdot \frac{100}{i}} \]

   18. Simplified 95.4%

       \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i \cdot \frac{0.01}{n}} - n \cdot \frac{100}{i}} \]

   if -2.0000000019e-315 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 22.2%

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

      1. associate-/r/ 22.2%

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

      2. sub-neg 22.2%

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

      3. metadata-eval 22.2%

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

   3. Simplified 22.2%

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

   5. Taylor expanded in n around inf 38.2%

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

      1. expm1-define 76.1%

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

   7. Simplified 76.1%

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

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

   1. Initial program 96.7%

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

      1. associate-*r/ 96.6%

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

      2. sub-neg 96.6%

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

      3. distribute-lft-in 96.5%

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

      4. metadata-eval 96.5%

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

      5. metadata-eval 96.5%

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

      6. metadata-eval 96.5%

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

      7. fma-define 96.6%

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

      8. metadata-eval 96.6%

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

   3. Simplified 96.6%

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

      1. fma-undefine 96.5%

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

      2. metadata-eval 96.5%

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

      3. distribute-lft-in 96.6%

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

      4. metadata-eval 96.6%

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

      5. sub-neg 96.6%

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

      6. *-commutative 96.6%

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

      7. add-exp-log 96.6%

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

      8. expm1-define 96.6%

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

      9. log-pow 52.8%

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

      10. log1p-define 52.8%

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

   6. Applied egg-rr 52.8%

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

      1. clear-num 52.8%

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

      2. inv-pow 52.8%

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

      3. *-commutative 52.8%

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

      4. associate-/r* 52.8%

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

   8. Applied egg-rr 52.8%

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

      1. unpow-1 52.8%

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

      2. associate-/l/ 52.8%

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

      3. *-commutative 52.8%

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

   10. Simplified 52.8%

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

       1. associate-/r* 52.8%

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

       2. associate-/r/ 52.9%

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

       3. associate-/l/ 52.9%

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

       4. *-commutative 52.9%

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

   12. Applied egg-rr 52.9%

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

       1. associate-*l/ 52.8%

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

       2. *-lft-identity 52.8%

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

   14. Simplified 52.8%

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

       1. expm1-undefine 49.8%

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

       2. *-commutative 49.8%

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

       3. log1p-undefine 49.8%

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

       4. pow-to-exp 96.5%

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

       5. div-sub 96.5%

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

       6. +-commutative 96.5%

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

       7. div-inv 96.5%

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

       8. *-commutative 96.5%

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

       9. associate-/r* 96.5%

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

       10. metadata-eval 96.5%

           \[\leadsto \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i \cdot \frac{\color{blue}{0.01}}{n}} - \frac{1}{\frac{i}{n \cdot 100}} \]

       11. clear-num 96.7%

           \[\leadsto \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i \cdot \frac{0.01}{n}} - \color{blue}{\frac{n \cdot 100}{i}} \]

   16. Applied egg-rr 96.7%

       \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i \cdot \frac{0.01}{n}} - \frac{n \cdot 100}{i}} \]

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-define 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

      1. fma-undefine 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. sub-neg 0.0%

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

      6. *-commutative 0.0%

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

      7. add-exp-log 0.0%

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

      8. expm1-define 0.0%

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

      9. log-pow 0.0%

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

      10. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/r* 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. unpow-1 0.0%

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

      2. associate-/l/ 0.0%

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

      3. *-commutative 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in i around 0 99.5%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-315}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i \cdot \frac{0.01}{n}} - n \cdot \frac{100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i \cdot \frac{0.01}{n}} - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-100 + t\_0 \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 10^{-263}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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 -1e-119)
     (/ (+ -100.0 (* t_0 100.0)) (/ i n))
     (if (<= t_1 1e-263)
       (* 100.0 (* (expm1 (* n (log1p (/ i n)))) (/ n i)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ (+ (pow (/ i n) n) -1.0) i)))
         (/
          1.0
          (+
           (* 0.01 (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))
           (* 0.01 (/ 1.0 n)))))))))

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 <= -1e-119) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else if (t_1 <= 1e-263) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) * (n / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-119) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else if (t_1 <= 1e-263) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) * (n / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((Math.pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-119:
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n)
	elif t_1 <= 1e-263:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) * (n / i))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((math.pow((i / n), n) + -1.0) / i))
	else:
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)))
	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 <= -1e-119)
		tmp = Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / Float64(i / n));
	elseif (t_1 <= 1e-263)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(n / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(i / n) ^ n) + -1.0) / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))) + Float64(0.01 * Float64(1.0 / n))));
	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, -1e-119], N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-263], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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)) < -1.00000000000000001e-119

   1. Initial program 99.4%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. fma-define 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -1.00000000000000001e-119 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-263

   1. Initial program 23.7%

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

      1. clear-num 23.7%

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

      2. associate-/r/ 23.6%

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

      3. clear-num 23.6%

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

      4. add-exp-log 23.6%

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

      5. expm1-define 23.6%

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

      6. log-pow 34.9%

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

      7. log1p-define 96.5%

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

   4. Applied egg-rr 96.5%

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

   if 1e-263 < (/.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%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. sub-neg 99.7%

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

      6. *-commutative 99.7%

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

      7. add-exp-log 99.7%

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

      8. expm1-define 99.7%

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

      9. log-pow 49.9%

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

      10. log1p-define 49.9%

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

   6. Applied egg-rr 49.9%

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

      1. clear-num 50.0%

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

      2. inv-pow 50.0%

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

      3. *-commutative 50.0%

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

      4. associate-/r* 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow-1 50.0%

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

      2. associate-/l/ 50.0%

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

      3. *-commutative 50.0%

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

   10. Simplified 50.0%

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

   11. Taylor expanded in i around inf 37.5%

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

       1. associate-/l* 37.5%

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

   13. Simplified 99.8%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-define 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

      1. fma-undefine 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. sub-neg 0.0%

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

      6. *-commutative 0.0%

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

      7. add-exp-log 0.0%

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

      8. expm1-define 0.0%

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

      9. log-pow 0.0%

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

      10. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/r* 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. unpow-1 0.0%

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

      2. associate-/l/ 0.0%

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

      3. *-commutative 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in i around 0 99.5%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-263}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-281}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{t\_0}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-263}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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-281)
     (* n (+ (* 100.0 (/ t_0 i)) (/ -100.0 i)))
     (if (<= t_1 1e-263)
       (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 (/ i n)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ (+ (pow (/ i n) n) -1.0) i)))
         (/
          1.0
          (+
           (* 0.01 (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))
           (* 0.01 (/ 1.0 n)))))))))

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-281) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = expm1((n * log1p((i / n)))) * (100.0 / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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-281) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) * (100.0 / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((Math.pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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-281:
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i))
	elif t_1 <= 1e-263:
		tmp = math.expm1((n * math.log1p((i / n)))) * (100.0 / (i / n))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((math.pow((i / n), n) + -1.0) / i))
	else:
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)))
	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-281)
		tmp = Float64(n * Float64(Float64(100.0 * Float64(t_0 / i)) + Float64(-100.0 / i)));
	elseif (t_1 <= 1e-263)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(i / n) ^ n) + -1.0) / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))) + Float64(0.01 * Float64(1.0 / n))));
	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-281], N[(n * N[(N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] + N[(-100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-263], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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-281

   1. Initial program 99.4%

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

      1. associate-*r/ 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.6%

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

      4. metadata-eval 99.6%

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

      5. sub-neg 99.6%

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

      6. *-commutative 99.6%

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

      7. add-exp-log 99.6%

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

      8. expm1-define 99.6%

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

      9. log-pow 63.2%

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

      10. log1p-define 63.2%

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

   6. Applied egg-rr 63.2%

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

      1. clear-num 63.4%

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

      2. inv-pow 63.4%

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

      3. *-commutative 63.4%

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

      4. associate-/r* 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. unpow-1 63.4%

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

      2. associate-/l/ 63.4%

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

      3. *-commutative 63.4%

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

   10. Simplified 63.4%

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

       1. associate-/r* 63.4%

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

       2. associate-/r/ 63.4%

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

       3. associate-/l/ 63.1%

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

       4. *-commutative 63.1%

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

   12. Applied egg-rr 63.1%

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

       1. associate-*l/ 63.1%

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

       2. *-lft-identity 63.1%

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

   14. Simplified 63.1%

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

       1. associate-/r/ 63.2%

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

       2. expm1-undefine 63.2%

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

       3. *-commutative 63.2%

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

       4. log1p-undefine 63.2%

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

       5. pow-to-exp 99.6%

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

       6. sub-div 99.6%

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

       7. *-commutative 99.6%

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

       8. sub-neg 99.6%

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

       9. distribute-lft-in 99.4%

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

       10. +-commutative 99.4%

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

       11. distribute-neg-frac 99.4%

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

       12. metadata-eval 99.4%

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

   16. Applied egg-rr 99.4%

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

       1. associate-*l* 99.4%

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

       2. associate-*l* 99.6%

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

       3. distribute-lft-out 99.8%

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

       4. +-commutative 99.8%

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

       5. associate-*r/ 100.0%

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

       6. metadata-eval 100.0%

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

   18. Simplified 100.0%

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

   if -2e-281 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-263

   1. Initial program 22.5%

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

      1. associate-*r/ 22.5%

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

      2. sub-neg 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. metadata-eval 22.5%

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

      5. metadata-eval 22.5%

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

      6. metadata-eval 22.5%

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

      7. fma-define 22.5%

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

      8. metadata-eval 22.5%

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

   3. Simplified 22.5%

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

      1. fma-undefine 22.5%

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

      2. metadata-eval 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. metadata-eval 22.5%

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

      5. sub-neg 22.5%

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

      6. *-commutative 22.5%

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

      7. add-exp-log 22.5%

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

      8. expm1-define 22.5%

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

      9. log-pow 34.1%

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

      10. log1p-define 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. clear-num 98.4%

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

      2. inv-pow 98.4%

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

      3. *-commutative 98.4%

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

      4. associate-/r* 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. unpow-1 96.3%

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

      2. associate-/l/ 98.4%

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

      3. *-commutative 98.4%

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

   10. Simplified 98.4%

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

       1. clear-num 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-/l* 96.6%

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

   12. Applied egg-rr 96.6%

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

   if 1e-263 < (/.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%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. sub-neg 99.7%

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

      6. *-commutative 99.7%

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

      7. add-exp-log 99.7%

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

      8. expm1-define 99.7%

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

      9. log-pow 49.9%

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

      10. log1p-define 49.9%

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

   6. Applied egg-rr 49.9%

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

      1. clear-num 50.0%

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

      2. inv-pow 50.0%

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

      3. *-commutative 50.0%

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

      4. associate-/r* 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow-1 50.0%

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

      2. associate-/l/ 50.0%

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

      3. *-commutative 50.0%

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

   10. Simplified 50.0%

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

   11. Taylor expanded in i around inf 37.5%

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

       1. associate-/l* 37.5%

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

   13. Simplified 99.8%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-define 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

      1. fma-undefine 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. sub-neg 0.0%

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

      6. *-commutative 0.0%

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

      7. add-exp-log 0.0%

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

      8. expm1-define 0.0%

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

      9. log-pow 0.0%

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

      10. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/r* 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. unpow-1 0.0%

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

      2. associate-/l/ 0.0%

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

      3. *-commutative 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in i around 0 99.5%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-263}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-247}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{t\_0}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-263}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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 -1e-247)
     (* n (+ (* 100.0 (/ t_0 i)) (/ -100.0 i)))
     (if (<= t_1 1e-263)
       (* (expm1 (* n (log1p (/ i n)))) (/ (* n 100.0) i))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ (+ (pow (/ i n) n) -1.0) i)))
         (/
          1.0
          (+
           (* 0.01 (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))
           (* 0.01 (/ 1.0 n)))))))))

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 <= -1e-247) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = expm1((n * log1p((i / n)))) * ((n * 100.0) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-247) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) * ((n * 100.0) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((Math.pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-247:
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i))
	elif t_1 <= 1e-263:
		tmp = math.expm1((n * math.log1p((i / n)))) * ((n * 100.0) / i)
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((math.pow((i / n), n) + -1.0) / i))
	else:
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)))
	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 <= -1e-247)
		tmp = Float64(n * Float64(Float64(100.0 * Float64(t_0 / i)) + Float64(-100.0 / i)));
	elseif (t_1 <= 1e-263)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(Float64(n * 100.0) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(i / n) ^ n) + -1.0) / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))) + Float64(0.01 * Float64(1.0 / n))));
	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, -1e-247], N[(n * N[(N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] + N[(-100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-263], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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)) < -1e-247

   1. Initial program 99.4%

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

      1. associate-*r/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

      7. fma-define 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. sub-neg 99.7%

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

      6. *-commutative 99.7%

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

      7. add-exp-log 99.7%

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

      8. expm1-define 99.7%

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

      9. log-pow 59.7%

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

      10. log1p-define 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. clear-num 59.7%

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

      2. inv-pow 59.7%

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

      3. *-commutative 59.7%

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

      4. associate-/r* 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. unpow-1 59.7%

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

      2. associate-/l/ 59.7%

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

      3. *-commutative 59.7%

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

   10. Simplified 59.7%

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

       1. associate-/r* 59.7%

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

       2. associate-/r/ 59.7%

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

       3. associate-/l/ 59.4%

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

       4. *-commutative 59.4%

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

   12. Applied egg-rr 59.4%

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

       1. associate-*l/ 59.4%

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

       2. *-lft-identity 59.4%

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

   14. Simplified 59.4%

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

       1. associate-/r/ 59.5%

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

       2. expm1-undefine 59.5%

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

       3. *-commutative 59.5%

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

       4. log1p-undefine 59.5%

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

       5. pow-to-exp 99.5%

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

       6. sub-div 99.5%

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

       7. *-commutative 99.5%

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

       8. sub-neg 99.5%

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

       9. distribute-lft-in 99.4%

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

       10. +-commutative 99.4%

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

       11. distribute-neg-frac 99.4%

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

       12. metadata-eval 99.4%

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

   16. Applied egg-rr 99.4%

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

       1. associate-*l* 99.4%

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

       2. associate-*l* 99.6%

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

       3. distribute-lft-out 99.8%

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

       4. +-commutative 99.8%

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

       5. associate-*r/ 100.0%

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

       6. metadata-eval 100.0%

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

   18. Simplified 100.0%

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

   if -1e-247 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-263

   1. Initial program 22.9%

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

      1. associate-*r/ 22.9%

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

      2. sub-neg 22.9%

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

      3. distribute-lft-in 22.9%

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

      4. metadata-eval 22.9%

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

      5. metadata-eval 22.9%

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

      6. metadata-eval 22.9%

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

      7. fma-define 22.9%

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

      8. metadata-eval 22.9%

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

   3. Simplified 22.9%

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

      1. associate-/r/ 22.9%

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

      2. associate-*l/ 22.9%

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

      3. fma-undefine 22.9%

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

      4. metadata-eval 22.9%

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

      5. distribute-lft-in 22.9%

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

      6. metadata-eval 22.9%

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

      7. sub-neg 22.9%

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

      8. *-commutative 22.9%

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

      9. associate-*r* 22.9%

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

      10. *-commutative 22.9%

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

      11. associate-/l* 22.8%

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

      12. add-exp-log 22.8%

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

      13. expm1-define 22.8%

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

      14. log-pow 34.3%

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

      15. log1p-define 96.7%

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

   6. Applied egg-rr 96.7%

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

   if 1e-263 < (/.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%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. sub-neg 99.7%

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

      6. *-commutative 99.7%

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

      7. add-exp-log 99.7%

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

      8. expm1-define 99.7%

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

      9. log-pow 49.9%

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

      10. log1p-define 49.9%

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

   6. Applied egg-rr 49.9%

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

      1. clear-num 50.0%

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

      2. inv-pow 50.0%

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

      3. *-commutative 50.0%

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

      4. associate-/r* 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow-1 50.0%

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

      2. associate-/l/ 50.0%

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

      3. *-commutative 50.0%

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

   10. Simplified 50.0%

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

   11. Taylor expanded in i around inf 37.5%

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

       1. associate-/l* 37.5%

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

   13. Simplified 99.8%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-define 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

      1. fma-undefine 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. sub-neg 0.0%

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

      6. *-commutative 0.0%

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

      7. add-exp-log 0.0%

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

      8. expm1-define 0.0%

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

      9. log-pow 0.0%

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

      10. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/r* 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. unpow-1 0.0%

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

      2. associate-/l/ 0.0%

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

      3. *-commutative 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in i around 0 99.5%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-247}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-263}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-247}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{t\_0}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-263}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n} \cdot 0.01}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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 -1e-247)
     (* n (+ (* 100.0 (/ t_0 i)) (/ -100.0 i)))
     (if (<= t_1 1e-263)
       (/ (expm1 (* n (log1p (/ i n)))) (* (/ i n) 0.01))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ (+ (pow (/ i n) n) -1.0) i)))
         (/
          1.0
          (+
           (* 0.01 (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))
           (* 0.01 (/ 1.0 n)))))))))

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 <= -1e-247) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = expm1((n * log1p((i / n)))) / ((i / n) * 0.01);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-247) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) / ((i / n) * 0.01);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((Math.pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-247:
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i))
	elif t_1 <= 1e-263:
		tmp = math.expm1((n * math.log1p((i / n)))) / ((i / n) * 0.01)
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((math.pow((i / n), n) + -1.0) / i))
	else:
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)))
	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 <= -1e-247)
		tmp = Float64(n * Float64(Float64(100.0 * Float64(t_0 / i)) + Float64(-100.0 / i)));
	elseif (t_1 <= 1e-263)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(Float64(i / n) * 0.01));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(i / n) ^ n) + -1.0) / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))) + Float64(0.01 * Float64(1.0 / n))));
	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, -1e-247], N[(n * N[(N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] + N[(-100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-263], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(N[(i / n), $MachinePrecision] * 0.01), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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)) < -1e-247

   1. Initial program 99.4%

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

      1. associate-*r/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

      7. fma-define 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. sub-neg 99.7%

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

      6. *-commutative 99.7%

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

      7. add-exp-log 99.7%

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

      8. expm1-define 99.7%

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

      9. log-pow 59.7%

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

      10. log1p-define 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. clear-num 59.7%

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

      2. inv-pow 59.7%

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

      3. *-commutative 59.7%

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

      4. associate-/r* 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. unpow-1 59.7%

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

      2. associate-/l/ 59.7%

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

      3. *-commutative 59.7%

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

   10. Simplified 59.7%

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

       1. associate-/r* 59.7%

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

       2. associate-/r/ 59.7%

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

       3. associate-/l/ 59.4%

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

       4. *-commutative 59.4%

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

   12. Applied egg-rr 59.4%

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

       1. associate-*l/ 59.4%

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

       2. *-lft-identity 59.4%

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

   14. Simplified 59.4%

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

       1. associate-/r/ 59.5%

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

       2. expm1-undefine 59.5%

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

       3. *-commutative 59.5%

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

       4. log1p-undefine 59.5%

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

       5. pow-to-exp 99.5%

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

       6. sub-div 99.5%

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

       7. *-commutative 99.5%

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

       8. sub-neg 99.5%

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

       9. distribute-lft-in 99.4%

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

       10. +-commutative 99.4%

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

       11. distribute-neg-frac 99.4%

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

       12. metadata-eval 99.4%

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

   16. Applied egg-rr 99.4%

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

       1. associate-*l* 99.4%

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

       2. associate-*l* 99.6%

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

       3. distribute-lft-out 99.8%

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

       4. +-commutative 99.8%

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

       5. associate-*r/ 100.0%

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

       6. metadata-eval 100.0%

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

   18. Simplified 100.0%

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

   if -1e-247 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-263

   1. Initial program 22.9%

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

      1. associate-*r/ 22.9%

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

      2. sub-neg 22.9%

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

      3. distribute-lft-in 22.9%

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

      4. metadata-eval 22.9%

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

      5. metadata-eval 22.9%

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

      6. metadata-eval 22.9%

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

      7. fma-define 22.9%

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

      8. metadata-eval 22.9%

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

   3. Simplified 22.9%

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

      1. fma-undefine 22.9%

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

      2. metadata-eval 22.9%

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

      3. distribute-lft-in 22.9%

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

      4. metadata-eval 22.9%

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

      5. sub-neg 22.9%

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

      6. *-commutative 22.9%

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

      7. add-exp-log 22.9%

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

      8. expm1-define 22.9%

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

      9. log-pow 34.4%

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

      10. log1p-define 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. clear-num 98.4%

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

      2. inv-pow 98.4%

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

      3. *-commutative 98.4%

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

      4. associate-/r* 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. unpow-1 96.3%

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

      2. associate-/l/ 98.4%

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

      3. *-commutative 98.4%

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

   10. Simplified 98.4%

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

       1. associate-/r* 96.3%

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

       2. associate-/r/ 96.5%

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

       3. associate-/l/ 96.6%

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

       4. *-commutative 96.6%

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

   12. Applied egg-rr 96.6%

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

       1. associate-*l/ 97.6%

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

       2. *-lft-identity 97.6%

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

   14. Simplified 97.6%

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

       1. associate-/r* 97.5%

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

       2. div-inv 97.5%

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

       3. metadata-eval 97.5%

          \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n} \cdot \color{blue}{0.01}} \]

   16. Applied egg-rr 97.5%

       \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{i}{n} \cdot 0.01}} \]

   if 1e-263 < (/.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%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. sub-neg 99.7%

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

      6. *-commutative 99.7%

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

      7. add-exp-log 99.7%

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

      8. expm1-define 99.7%

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

      9. log-pow 49.9%

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

      10. log1p-define 49.9%

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

   6. Applied egg-rr 49.9%

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

      1. clear-num 50.0%

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

      2. inv-pow 50.0%

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

      3. *-commutative 50.0%

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

      4. associate-/r* 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow-1 50.0%

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

      2. associate-/l/ 50.0%

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

      3. *-commutative 50.0%

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

   10. Simplified 50.0%

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

   11. Taylor expanded in i around inf 37.5%

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

       1. associate-/l* 37.5%

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

   13. Simplified 99.8%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 0.0%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{100 \cdot -1}}{\frac{i}{n}} \]

      3. distribute-lft-in 0.0%

         \[\leadsto \frac{\color{blue}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}\right)}{\frac{i}{n}} \]

      5. sub-neg 0.0%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      6. *-commutative 0.0%

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]

      7. add-exp-log 0.0%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot 100}{\frac{i}{n}} \]

      8. expm1-define 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot 100}{\frac{i}{n}} \]

      9. log-pow 0.0%

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

      10. log1p-define 0.0%

          \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{\frac{i}{n}} \]
   7. Step-by-step derivation

      1. clear-num 0.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      2. inv-pow 0.0%

         \[\leadsto \color{blue}{{\left(\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}\right)}^{-1}} \]

      3. *-commutative 0.0%

         \[\leadsto {\left(\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\right)}^{-1} \]

      4. associate-/r* 0.0%

         \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}}^{-1} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 0.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. associate-/l/ 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{1}{\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   11. Taylor expanded in i around 0 99.5%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-247}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-263}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n} \cdot 0.01}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-247}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{t\_0}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-263}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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 -1e-247)
     (* n (+ (* 100.0 (/ t_0 i)) (/ -100.0 i)))
     (if (<= t_1 1e-263)
       (/ (expm1 (* n (log1p (/ i n)))) (/ i (* n 100.0)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ (+ (pow (/ i n) n) -1.0) i)))
         (/
          1.0
          (+
           (* 0.01 (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))
           (* 0.01 (/ 1.0 n)))))))))

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 <= -1e-247) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = expm1((n * log1p((i / n)))) / (i / (n * 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-247) {
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i));
	} else if (t_1 <= 1e-263) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) / (i / (n * 100.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((Math.pow((i / n), n) + -1.0) / i));
	} else {
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)));
	}
	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 <= -1e-247:
		tmp = n * ((100.0 * (t_0 / i)) + (-100.0 / i))
	elif t_1 <= 1e-263:
		tmp = math.expm1((n * math.log1p((i / n)))) / (i / (n * 100.0))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((math.pow((i / n), n) + -1.0) / i))
	else:
		tmp = 1.0 / ((0.01 * (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))) + (0.01 * (1.0 / n)))
	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 <= -1e-247)
		tmp = Float64(n * Float64(Float64(100.0 * Float64(t_0 / i)) + Float64(-100.0 / i)));
	elseif (t_1 <= 1e-263)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / Float64(n * 100.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(i / n) ^ n) + -1.0) / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))) + Float64(0.01 * Float64(1.0 / n))));
	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, -1e-247], N[(n * N[(N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] + N[(-100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-263], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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)) < -1e-247

   1. Initial program 99.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 99.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 99.8%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 99.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.8%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{100 \cdot -1}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.7%

         \[\leadsto \frac{\color{blue}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}\right)}{\frac{i}{n}} \]

      5. sub-neg 99.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      6. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]

      7. add-exp-log 99.7%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot 100}{\frac{i}{n}} \]

      8. expm1-define 99.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot 100}{\frac{i}{n}} \]

      9. log-pow 59.7%

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

      10. log1p-define 59.7%

          \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   6. Applied egg-rr 59.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{\frac{i}{n}} \]
   7. Step-by-step derivation

      1. clear-num 59.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      2. inv-pow 59.7%

         \[\leadsto \color{blue}{{\left(\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}\right)}^{-1}} \]

      3. *-commutative 59.7%

         \[\leadsto {\left(\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\right)}^{-1} \]

      4. associate-/r* 59.7%

         \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}}^{-1} \]

   8. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 59.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. associate-/l/ 59.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      3. *-commutative 59.7%

         \[\leadsto \frac{1}{\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   10. Simplified 59.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]
   11. Step-by-step derivation

       1. associate-/r* 59.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

       2. associate-/r/ 59.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)} \]

       3. associate-/l/ 59.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{i}{100 \cdot n}}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \]

       4. *-commutative 59.4%

          \[\leadsto \frac{1}{\frac{i}{\color{blue}{n \cdot 100}}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \]

   12. Applied egg-rr 59.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{i}{n \cdot 100}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 59.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}} \]

       2. *-lft-identity 59.4%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n \cdot 100}} \]

   14. Simplified 59.4%

       \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}} \]
   15. Step-by-step derivation

       1. associate-/r/ 59.5%

          \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)} \]

       2. expm1-undefine 59.5%

          \[\leadsto \frac{\color{blue}{e^{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)} - 1}}{i} \cdot \left(n \cdot 100\right) \]

       3. *-commutative 59.5%

          \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1}{i} \cdot \left(n \cdot 100\right) \]

       4. log1p-undefine 59.5%

          \[\leadsto \frac{e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1}{i} \cdot \left(n \cdot 100\right) \]

       5. pow-to-exp 99.5%

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{i} \cdot \left(n \cdot 100\right) \]

       6. sub-div 99.5%

          \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{1}{i}\right)} \cdot \left(n \cdot 100\right) \]

       7. *-commutative 99.5%

          \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{1}{i}\right)} \]

       8. sub-neg 99.5%

          \[\leadsto \left(n \cdot 100\right) \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \left(-\frac{1}{i}\right)\right)} \]

       9. distribute-lft-in 99.4%

          \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \left(n \cdot 100\right) \cdot \left(-\frac{1}{i}\right)} \]

       10. +-commutative 99.4%

           \[\leadsto \left(n \cdot 100\right) \cdot \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} + \left(n \cdot 100\right) \cdot \left(-\frac{1}{i}\right) \]

       11. distribute-neg-frac 99.4%

           \[\leadsto \left(n \cdot 100\right) \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} + \left(n \cdot 100\right) \cdot \color{blue}{\frac{-1}{i}} \]

       12. metadata-eval 99.4%

           \[\leadsto \left(n \cdot 100\right) \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} + \left(n \cdot 100\right) \cdot \frac{\color{blue}{-1}}{i} \]

   16. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} + \left(n \cdot 100\right) \cdot \frac{-1}{i}} \]
   17. Step-by-step derivation

       1. associate-*l* 99.4%

          \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}\right)} + \left(n \cdot 100\right) \cdot \frac{-1}{i} \]

       2. associate-*l* 99.6%

          \[\leadsto n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}\right) + \color{blue}{n \cdot \left(100 \cdot \frac{-1}{i}\right)} \]

       3. distribute-lft-out 99.8%

          \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} + 100 \cdot \frac{-1}{i}\right)} \]

       4. +-commutative 99.8%

          \[\leadsto n \cdot \left(100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} + 100 \cdot \frac{-1}{i}\right) \]

       5. associate-*r/ 100.0%

          \[\leadsto n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \color{blue}{\frac{100 \cdot -1}{i}}\right) \]

       6. metadata-eval 100.0%

          \[\leadsto n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{\color{blue}{-100}}{i}\right) \]

   18. Simplified 100.0%

       \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-100}{i}\right)} \]

   if -1e-247 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-263

   1. Initial program 22.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 22.9%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 22.9%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 22.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 22.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 22.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 22.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 22.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 22.9%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 22.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 22.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. metadata-eval 22.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{100 \cdot -1}}{\frac{i}{n}} \]

      3. distribute-lft-in 22.9%

         \[\leadsto \frac{\color{blue}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 22.9%

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}\right)}{\frac{i}{n}} \]

      5. sub-neg 22.9%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      6. *-commutative 22.9%

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]

      7. add-exp-log 22.9%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot 100}{\frac{i}{n}} \]

      8. expm1-define 22.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot 100}{\frac{i}{n}} \]

      9. log-pow 34.4%

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

      10. log1p-define 99.7%

          \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{\frac{i}{n}} \]
   7. Step-by-step derivation

      1. clear-num 98.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      2. inv-pow 98.4%

         \[\leadsto \color{blue}{{\left(\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}\right)}^{-1}} \]

      3. *-commutative 98.4%

         \[\leadsto {\left(\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\right)}^{-1} \]

      4. associate-/r* 96.3%

         \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}}^{-1} \]

   8. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 96.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. associate-/l/ 98.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      3. *-commutative 98.4%

         \[\leadsto \frac{1}{\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   10. Simplified 98.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]
   11. Step-by-step derivation

       1. associate-/r* 96.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

       2. associate-/r/ 96.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)} \]

       3. associate-/l/ 96.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{i}{100 \cdot n}}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \]

       4. *-commutative 96.6%

          \[\leadsto \frac{1}{\frac{i}{\color{blue}{n \cdot 100}}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \]

   12. Applied egg-rr 96.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{i}{n \cdot 100}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 97.6%

          \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}} \]

       2. *-lft-identity 97.6%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n \cdot 100}} \]

   14. Simplified 97.6%

       \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}} \]

   if 1e-263 < (/.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%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 99.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 99.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{100 \cdot -1}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.7%

         \[\leadsto \frac{\color{blue}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}\right)}{\frac{i}{n}} \]

      5. sub-neg 99.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      6. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]

      7. add-exp-log 99.7%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot 100}{\frac{i}{n}} \]

      8. expm1-define 99.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot 100}{\frac{i}{n}} \]

      9. log-pow 49.9%

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

      10. log1p-define 49.9%

          \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   6. Applied egg-rr 49.9%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{\frac{i}{n}} \]
   7. Step-by-step derivation

      1. clear-num 50.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      2. inv-pow 50.0%

         \[\leadsto \color{blue}{{\left(\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}\right)}^{-1}} \]

      3. *-commutative 50.0%

         \[\leadsto {\left(\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\right)}^{-1} \]

      4. associate-/r* 50.0%

         \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}}^{-1} \]

   8. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 50.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. associate-/l/ 50.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      3. *-commutative 50.0%

         \[\leadsto \frac{1}{\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   10. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   11. Taylor expanded in i around inf 37.5%

       \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{n \cdot \left(\log \left(\frac{1}{n}\right) + -1 \cdot \log \left(\frac{1}{i}\right)\right)} - 1\right)}{i}} \]
   12. Step-by-step derivation

       1. associate-/l* 37.5%

          \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{n \cdot \left(\log \left(\frac{1}{n}\right) + -1 \cdot \log \left(\frac{1}{i}\right)\right)} - 1}{i}\right)} \]

   13. Simplified 99.8%

       \[\leadsto \color{blue}{100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)} \]

   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/ 0.0%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 0.0%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 0.0%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 0.0%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 0.0%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{100 \cdot -1}}{\frac{i}{n}} \]

      3. distribute-lft-in 0.0%

         \[\leadsto \frac{\color{blue}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 0.0%

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}\right)}{\frac{i}{n}} \]

      5. sub-neg 0.0%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      6. *-commutative 0.0%

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{\frac{i}{n}} \]

      7. add-exp-log 0.0%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot 100}{\frac{i}{n}} \]

      8. expm1-define 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot 100}{\frac{i}{n}} \]

      9. log-pow 0.0%

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

      10. log1p-define 0.0%

          \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right) \cdot 100}{\frac{i}{n}} \]

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}{\frac{i}{n}} \]
   7. Step-by-step derivation

      1. clear-num 0.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      2. inv-pow 0.0%

         \[\leadsto \color{blue}{{\left(\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}\right)}^{-1}} \]

      3. *-commutative 0.0%

         \[\leadsto {\left(\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\right)}^{-1} \]

      4. associate-/r* 0.0%

         \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}}^{-1} \]

   8. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 0.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{i}{n}}{100}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. associate-/l/ 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{1}{\frac{\frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   11. Taylor expanded in i around 0 99.5%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-247}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-263}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.42 \cdot 10^{-238}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.9:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (/ (expm1 i) i)))))
   (if (<= n -1.42e-238)
     t_0
     (if (<= n 2.6e-218) 0.0 (if (<= n 2.9) (/ (* i 100.0) (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (expm1(i) / i));
	double tmp;
	if (n <= -1.42e-238) {
		tmp = t_0;
	} else if (n <= 2.6e-218) {
		tmp = 0.0;
	} else if (n <= 2.9) {
		tmp = (i * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n * (Math.expm1(i) / i));
	double tmp;
	if (n <= -1.42e-238) {
		tmp = t_0;
	} else if (n <= 2.6e-218) {
		tmp = 0.0;
	} else if (n <= 2.9) {
		tmp = (i * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (math.expm1(i) / i))
	tmp = 0
	if n <= -1.42e-238:
		tmp = t_0
	elif n <= 2.6e-218:
		tmp = 0.0
	elif n <= 2.9:
		tmp = (i * 100.0) / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.42e-238)
		tmp = t_0;
	elseif (n <= 2.6e-218)
		tmp = 0.0;
	elseif (n <= 2.9)
		tmp = Float64(Float64(i * 100.0) / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.42e-238], t$95$0, If[LessEqual[n, 2.6e-218], 0.0, If[LessEqual[n, 2.9], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.4199999999999999e-238 or 2.89999999999999991 < n

   1. Initial program 21.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 22.2%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. sub-neg 22.2%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \]

      3. metadata-eval 22.2%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \]

   3. Simplified 22.2%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 36.1%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{i} - 1}}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. expm1-define 84.4%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   7. Simplified 84.4%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   if -1.4199999999999999e-238 < n < 2.59999999999999983e-218

   1. Initial program 71.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 71.7%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 71.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 71.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 71.7%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 71.7%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 71.7%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   7. Taylor expanded in i around 0 80.4%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0 80.4%

      \[\leadsto \color{blue}{0} \]

   if 2.59999999999999983e-218 < n < 2.89999999999999991

   1. Initial program 23.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 23.6%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 23.6%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 23.5%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 23.5%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 23.5%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 23.5%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 23.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 23.6%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 23.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 66.7%

      \[\leadsto \frac{\color{blue}{100 \cdot i}}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]

   7. Simplified 66.7%

      \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.42 \cdot 10^{-238}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.9:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ t_1 := \frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -7 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-243}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-217}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.16 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))) (t_1 (/ (* i 100.0) (/ i n))))
   (if (<= n -7e+78)
     t_0
     (if (<= n -4.5e-243)
       t_1
       (if (<= n 4e-217) 0.0 (if (<= n 1.16e-10) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = (i * 100.0) / (i / n);
	double tmp;
	if (n <= -7e+78) {
		tmp = t_0;
	} else if (n <= -4.5e-243) {
		tmp = t_1;
	} else if (n <= 4e-217) {
		tmp = 0.0;
	} else if (n <= 1.16e-10) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    t_1 = (i * 100.0d0) / (i / n)
    if (n <= (-7d+78)) then
        tmp = t_0
    else if (n <= (-4.5d-243)) then
        tmp = t_1
    else if (n <= 4d-217) then
        tmp = 0.0d0
    else if (n <= 1.16d-10) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = (i * 100.0) / (i / n);
	double tmp;
	if (n <= -7e+78) {
		tmp = t_0;
	} else if (n <= -4.5e-243) {
		tmp = t_1;
	} else if (n <= 4e-217) {
		tmp = 0.0;
	} else if (n <= 1.16e-10) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	t_1 = (i * 100.0) / (i / n)
	tmp = 0
	if n <= -7e+78:
		tmp = t_0
	elif n <= -4.5e-243:
		tmp = t_1
	elif n <= 4e-217:
		tmp = 0.0
	elif n <= 1.16e-10:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	t_1 = Float64(Float64(i * 100.0) / Float64(i / n))
	tmp = 0.0
	if (n <= -7e+78)
		tmp = t_0;
	elseif (n <= -4.5e-243)
		tmp = t_1;
	elseif (n <= 4e-217)
		tmp = 0.0;
	elseif (n <= 1.16e-10)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	t_1 = (i * 100.0) / (i / n);
	tmp = 0.0;
	if (n <= -7e+78)
		tmp = t_0;
	elseif (n <= -4.5e-243)
		tmp = t_1;
	elseif (n <= 4e-217)
		tmp = 0.0;
	elseif (n <= 1.16e-10)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7e+78], t$95$0, If[LessEqual[n, -4.5e-243], t$95$1, If[LessEqual[n, 4e-217], 0.0, If[LessEqual[n, 1.16e-10], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.0000000000000003e78 or 1.16e-10 < n

   1. Initial program 16.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 17.0%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. sub-neg 17.0%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \]

      3. metadata-eval 17.0%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \]

   3. Simplified 17.0%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 41.0%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{i} - 1}}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. expm1-define 93.4%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   7. Simplified 93.4%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   8. Taylor expanded in i around 0 70.7%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \color{blue}{100 \cdot n + 50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* 70.7%

         \[\leadsto 100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n} \]

      3. distribute-rgt-out 70.7%

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   if -7.0000000000000003e78 < n < -4.50000000000000017e-243 or 4.00000000000000033e-217 < n < 1.16e-10

   1. Initial program 28.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 28.6%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 28.6%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 28.6%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 28.6%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 28.6%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 28.6%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 28.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 28.6%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 61.0%

      \[\leadsto \frac{\color{blue}{100 \cdot i}}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. *-commutative 61.0%

         \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]

   7. Simplified 61.0%

      \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]

   if -4.50000000000000017e-243 < n < 4.00000000000000033e-217

   1. Initial program 71.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 71.7%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 71.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 71.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 71.7%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 71.7%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 71.7%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   7. Taylor expanded in i around 0 80.4%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0 80.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{+78}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-243}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-217}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.16 \cdot 10^{-10}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -1.4 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-219}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.002:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* i 100.0) (/ i n))))
   (if (<= n -3.6e-30)
     (/ (* 100.0 (* i n)) i)
     (if (<= n -1.4e-242)
       t_0
       (if (<= n 9e-219)
         0.0
         (if (<= n 0.002) t_0 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = (i * 100.0) / (i / n);
	double tmp;
	if (n <= -3.6e-30) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -1.4e-242) {
		tmp = t_0;
	} else if (n <= 9e-219) {
		tmp = 0.0;
	} else if (n <= 0.002) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * 100.0d0) / (i / n)
    if (n <= (-3.6d-30)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= (-1.4d-242)) then
        tmp = t_0
    else if (n <= 9d-219) then
        tmp = 0.0d0
    else if (n <= 0.002d0) then
        tmp = t_0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (i * 100.0) / (i / n);
	double tmp;
	if (n <= -3.6e-30) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -1.4e-242) {
		tmp = t_0;
	} else if (n <= 9e-219) {
		tmp = 0.0;
	} else if (n <= 0.002) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i * 100.0) / (i / n)
	tmp = 0
	if n <= -3.6e-30:
		tmp = (100.0 * (i * n)) / i
	elif n <= -1.4e-242:
		tmp = t_0
	elif n <= 9e-219:
		tmp = 0.0
	elif n <= 0.002:
		tmp = t_0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i * 100.0) / Float64(i / n))
	tmp = 0.0
	if (n <= -3.6e-30)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= -1.4e-242)
		tmp = t_0;
	elseif (n <= 9e-219)
		tmp = 0.0;
	elseif (n <= 0.002)
		tmp = t_0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (i * 100.0) / (i / n);
	tmp = 0.0;
	if (n <= -3.6e-30)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= -1.4e-242)
		tmp = t_0;
	elseif (n <= 9e-219)
		tmp = 0.0;
	elseif (n <= 0.002)
		tmp = t_0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.6e-30], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -1.4e-242], t$95$0, If[LessEqual[n, 9e-219], 0.0, If[LessEqual[n, 0.002], t$95$0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.6000000000000003e-30

   1. Initial program 21.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 22.2%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. sub-neg 22.2%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \]

      3. metadata-eval 22.2%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \]

   3. Simplified 22.2%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 35.2%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{i} - 1}}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. expm1-define 87.4%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   7. Simplified 87.4%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]
   8. Step-by-step derivation

      1. *-commutative 87.4%

         \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100} \]

      2. associate-*l/ 87.4%

         \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(i\right) \cdot n}{i}} \cdot 100 \]

      3. associate-*l/ 87.2%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{expm1}\left(i\right) \cdot n\right) \cdot 100}{i}} \]

      4. *-commutative 87.2%

         \[\leadsto \frac{\color{blue}{\left(n \cdot \mathsf{expm1}\left(i\right)\right)} \cdot 100}{i} \]

   9. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\frac{\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot 100}{i}} \]

   10. Taylor expanded in i around 0 61.9%

       \[\leadsto \frac{\color{blue}{\left(i \cdot n\right)} \cdot 100}{i} \]
   11. Step-by-step derivation

       1. *-commutative 61.9%

          \[\leadsto \frac{\color{blue}{\left(n \cdot i\right)} \cdot 100}{i} \]

   12. Simplified 61.9%

       \[\leadsto \frac{\color{blue}{\left(n \cdot i\right)} \cdot 100}{i} \]

   if -3.6000000000000003e-30 < n < -1.39999999999999992e-242 or 9.00000000000000029e-219 < n < 2e-3

   1. Initial program 26.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 26.3%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 26.3%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 26.3%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 26.3%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 26.3%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 26.3%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 26.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 26.3%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 26.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 63.6%

      \[\leadsto \frac{\color{blue}{100 \cdot i}}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. *-commutative 63.6%

         \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]

   7. Simplified 63.6%

      \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]

   if -1.39999999999999992e-242 < n < 9.00000000000000029e-219

   1. Initial program 71.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 71.7%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 71.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 71.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 71.7%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 71.7%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 71.7%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   7. Taylor expanded in i around 0 80.4%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0 80.4%

      \[\leadsto \color{blue}{0} \]

   if 2e-3 < n

   1. Initial program 17.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 18.2%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. sub-neg 18.2%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \]

      3. metadata-eval 18.2%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \]

   3. Simplified 18.2%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 43.0%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{i} - 1}}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. expm1-define 94.4%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   7. Simplified 94.4%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   8. Taylor expanded in i around 0 73.9%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. Step-by-step derivation

      1. +-commutative 73.9%

         \[\leadsto \color{blue}{100 \cdot n + 50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* 73.9%

         \[\leadsto 100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n} \]

      3. distribute-rgt-out 73.9%

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   10. Simplified 73.9%

       \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -1.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-219}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.002:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -7 \cdot 10^{+78}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\frac{1}{i + 2} \cdot \left(i \cdot \left(i + 2\right)\right)}{i}\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{-244}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-218}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.053:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* i 100.0) (/ i n))))
   (if (<= n -7e+78)
     (* (* n 100.0) (/ (* (/ 1.0 (+ i 2.0)) (* i (+ i 2.0))) i))
     (if (<= n -6.4e-244)
       t_0
       (if (<= n 8e-218)
         0.0
         (if (<= n 0.053) t_0 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = (i * 100.0) / (i / n);
	double tmp;
	if (n <= -7e+78) {
		tmp = (n * 100.0) * (((1.0 / (i + 2.0)) * (i * (i + 2.0))) / i);
	} else if (n <= -6.4e-244) {
		tmp = t_0;
	} else if (n <= 8e-218) {
		tmp = 0.0;
	} else if (n <= 0.053) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * 100.0d0) / (i / n)
    if (n <= (-7d+78)) then
        tmp = (n * 100.0d0) * (((1.0d0 / (i + 2.0d0)) * (i * (i + 2.0d0))) / i)
    else if (n <= (-6.4d-244)) then
        tmp = t_0
    else if (n <= 8d-218) then
        tmp = 0.0d0
    else if (n <= 0.053d0) then
        tmp = t_0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (i * 100.0) / (i / n);
	double tmp;
	if (n <= -7e+78) {
		tmp = (n * 100.0) * (((1.0 / (i + 2.0)) * (i * (i + 2.0))) / i);
	} else if (n <= -6.4e-244) {
		tmp = t_0;
	} else if (n <= 8e-218) {
		tmp = 0.0;
	} else if (n <= 0.053) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i * 100.0) / (i / n)
	tmp = 0
	if n <= -7e+78:
		tmp = (n * 100.0) * (((1.0 / (i + 2.0)) * (i * (i + 2.0))) / i)
	elif n <= -6.4e-244:
		tmp = t_0
	elif n <= 8e-218:
		tmp = 0.0
	elif n <= 0.053:
		tmp = t_0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i * 100.0) / Float64(i / n))
	tmp = 0.0
	if (n <= -7e+78)
		tmp = Float64(Float64(n * 100.0) * Float64(Float64(Float64(1.0 / Float64(i + 2.0)) * Float64(i * Float64(i + 2.0))) / i));
	elseif (n <= -6.4e-244)
		tmp = t_0;
	elseif (n <= 8e-218)
		tmp = 0.0;
	elseif (n <= 0.053)
		tmp = t_0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (i * 100.0) / (i / n);
	tmp = 0.0;
	if (n <= -7e+78)
		tmp = (n * 100.0) * (((1.0 / (i + 2.0)) * (i * (i + 2.0))) / i);
	elseif (n <= -6.4e-244)
		tmp = t_0;
	elseif (n <= 8e-218)
		tmp = 0.0;
	elseif (n <= 0.053)
		tmp = t_0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7e+78], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(N[(1.0 / N[(i + 2.0), $MachinePrecision]), $MachinePrecision] * N[(i * N[(i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -6.4e-244], t$95$0, If[LessEqual[n, 8e-218], 0.0, If[LessEqual[n, 0.053], t$95$0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -7.0000000000000003e78

   1. Initial program 14.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 14.9%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 15.3%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 15.3%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 15.3%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 15.3%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 15.3%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 4.3%

      \[\leadsto \frac{\color{blue}{\left(1 + i\right)} + -1}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative 4.3%

         \[\leadsto \frac{\color{blue}{\left(i + 1\right)} + -1}{i} \cdot \left(n \cdot 100\right) \]

   7. Simplified 4.3%

      \[\leadsto \frac{\color{blue}{\left(i + 1\right)} + -1}{i} \cdot \left(n \cdot 100\right) \]
   8. Step-by-step derivation

      1. flip-+ 13.1%

         \[\leadsto \frac{\color{blue}{\frac{\left(i + 1\right) \cdot \left(i + 1\right) - -1 \cdot -1}{\left(i + 1\right) - -1}}}{i} \cdot \left(n \cdot 100\right) \]

      2. div-inv 13.1%

         \[\leadsto \frac{\color{blue}{\left(\left(i + 1\right) \cdot \left(i + 1\right) - -1 \cdot -1\right) \cdot \frac{1}{\left(i + 1\right) - -1}}}{i} \cdot \left(n \cdot 100\right) \]

      3. metadata-eval 13.1%

         \[\leadsto \frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - \color{blue}{1}\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      4. difference-of-sqr-1 13.1%

         \[\leadsto \frac{\color{blue}{\left(\left(\left(i + 1\right) + 1\right) \cdot \left(\left(i + 1\right) - 1\right)\right)} \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 13.1%

         \[\leadsto \frac{\left(\left(\left(i + 1\right) + \color{blue}{\left(--1\right)}\right) \cdot \left(\left(i + 1\right) - 1\right)\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      6. sub-neg 13.1%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(i + 1\right) - -1\right)} \cdot \left(\left(i + 1\right) - 1\right)\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      7. associate--l+ 13.1%

         \[\leadsto \frac{\left(\color{blue}{\left(i + \left(1 - -1\right)\right)} \cdot \left(\left(i + 1\right) - 1\right)\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      8. metadata-eval 13.1%

         \[\leadsto \frac{\left(\left(i + \color{blue}{2}\right) \cdot \left(\left(i + 1\right) - 1\right)\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      9. add-exp-log 11.8%

         \[\leadsto \frac{\left(\left(i + 2\right) \cdot \left(\color{blue}{e^{\log \left(i + 1\right)}} - 1\right)\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      10. expm1-undefine 11.8%

          \[\leadsto \frac{\left(\left(i + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \left(i + 1\right)\right)}\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      11. +-commutative 11.8%

          \[\leadsto \frac{\left(\left(i + 2\right) \cdot \mathsf{expm1}\left(\log \color{blue}{\left(1 + i\right)}\right)\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      12. log1p-define 65.6%

          \[\leadsto \frac{\left(\left(i + 2\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(i\right)}\right)\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      13. expm1-log1p-u 66.9%

          \[\leadsto \frac{\left(\left(i + 2\right) \cdot \color{blue}{i}\right) \cdot \frac{1}{\left(i + 1\right) - -1}}{i} \cdot \left(n \cdot 100\right) \]

      14. associate--l+ 66.9%

          \[\leadsto \frac{\left(\left(i + 2\right) \cdot i\right) \cdot \frac{1}{\color{blue}{i + \left(1 - -1\right)}}}{i} \cdot \left(n \cdot 100\right) \]

      15. metadata-eval 66.9%

          \[\leadsto \frac{\left(\left(i + 2\right) \cdot i\right) \cdot \frac{1}{i + \color{blue}{2}}}{i} \cdot \left(n \cdot 100\right) \]

   9. Applied egg-rr 66.9%

      \[\leadsto \frac{\color{blue}{\left(\left(i + 2\right) \cdot i\right) \cdot \frac{1}{i + 2}}}{i} \cdot \left(n \cdot 100\right) \]
   10. Step-by-step derivation

       1. *-commutative 66.9%

          \[\leadsto \frac{\color{blue}{\frac{1}{i + 2} \cdot \left(\left(i + 2\right) \cdot i\right)}}{i} \cdot \left(n \cdot 100\right) \]

       2. *-commutative 66.9%

          \[\leadsto \frac{\frac{1}{i + 2} \cdot \color{blue}{\left(i \cdot \left(i + 2\right)\right)}}{i} \cdot \left(n \cdot 100\right) \]

   11. Simplified 66.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{i + 2} \cdot \left(i \cdot \left(i + 2\right)\right)}}{i} \cdot \left(n \cdot 100\right) \]

   if -7.0000000000000003e78 < n < -6.3999999999999996e-244 or 8.0000000000000003e-218 < n < 0.0529999999999999985

   1. Initial program 28.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 28.6%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 28.6%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 28.6%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 28.6%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 28.6%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 28.6%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 28.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 28.6%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 61.0%

      \[\leadsto \frac{\color{blue}{100 \cdot i}}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. *-commutative 61.0%

         \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]

   7. Simplified 61.0%

      \[\leadsto \frac{\color{blue}{i \cdot 100}}{\frac{i}{n}} \]

   if -6.3999999999999996e-244 < n < 8.0000000000000003e-218

   1. Initial program 71.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 71.7%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 71.7%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 71.7%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 71.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 71.7%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 71.7%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 71.7%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 71.7%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   7. Taylor expanded in i around 0 80.4%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0 80.4%

      \[\leadsto \color{blue}{0} \]

   if 0.0529999999999999985 < n

   1. Initial program 17.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 18.2%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. sub-neg 18.2%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \]

      3. metadata-eval 18.2%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \]

   3. Simplified 18.2%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 43.0%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{i} - 1}}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. expm1-define 94.4%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   7. Simplified 94.4%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   8. Taylor expanded in i around 0 73.9%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. Step-by-step derivation

      1. +-commutative 73.9%

         \[\leadsto \color{blue}{100 \cdot n + 50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* 73.9%

         \[\leadsto 100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n} \]

      3. distribute-rgt-out 73.9%

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   10. Simplified 73.9%

       \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{+78}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\frac{1}{i + 2} \cdot \left(i \cdot \left(i + 2\right)\right)}{i}\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{-244}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-218}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.053:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-169} \lor \neg \left(n \leq 4.6 \cdot 10^{-42}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.1e-169) (not (<= n 4.6e-42)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.1e-169) || !(n <= 4.6e-42)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-4.1d-169)) .or. (.not. (n <= 4.6d-42))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.1e-169) || !(n <= 4.6e-42)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.1e-169) or not (n <= 4.6e-42):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.1e-169) || !(n <= 4.6e-42))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -4.1e-169) || ~((n <= 4.6e-42)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.1e-169], N[Not[LessEqual[n, 4.6e-42]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -4.0999999999999998e-169 or 4.60000000000000008e-42 < n

   1. Initial program 19.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 19.7%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. sub-neg 19.7%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \]

      3. metadata-eval 19.7%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \]

   3. Simplified 19.7%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 34.8%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{i} - 1}}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. expm1-define 85.7%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   7. Simplified 85.7%

      \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot n\right) \]

   8. Taylor expanded in i around 0 64.6%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. Step-by-step derivation

      1. +-commutative 64.6%

         \[\leadsto \color{blue}{100 \cdot n + 50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* 64.6%

         \[\leadsto 100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n} \]

      3. distribute-rgt-out 64.6%

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   10. Simplified 64.6%

       \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

   if -4.0999999999999998e-169 < n < 4.60000000000000008e-42

   1. Initial program 42.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 42.5%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 42.5%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 42.5%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 42.5%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 42.5%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 42.5%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 42.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 42.5%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 42.5%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 42.5%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 42.5%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   7. Taylor expanded in i around 0 59.8%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0 59.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-169} \lor \neg \left(n \leq 4.6 \cdot 10^{-42}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -122000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 28000000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -122000000000.0) 0.0 (if (<= i 28000000000.0) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -122000000000.0) {
		tmp = 0.0;
	} else if (i <= 28000000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-122000000000.0d0)) then
        tmp = 0.0d0
    else if (i <= 28000000000.0d0) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -122000000000.0) {
		tmp = 0.0;
	} else if (i <= 28000000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -122000000000.0:
		tmp = 0.0
	elif i <= 28000000000.0:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -122000000000.0)
		tmp = 0.0;
	elseif (i <= 28000000000.0)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -122000000000.0)
		tmp = 0.0;
	elseif (i <= 28000000000.0)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -122000000000.0], 0.0, If[LessEqual[i, 28000000000.0], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.22e11 or 2.8e10 < i

   1. Initial program 48.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 48.8%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 48.8%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 48.8%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 48.8%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 48.8%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 48.8%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-define 48.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 48.8%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 48.8%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 48.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 48.8%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   7. Taylor expanded in i around 0 36.9%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0 36.9%

      \[\leadsto \color{blue}{0} \]

   if -1.22e11 < i < 2.8e10

   1. Initial program 11.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 11.9%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. sub-neg 11.9%

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \]

      3. metadata-eval 11.9%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \]

   3. Simplified 11.9%

      \[\leadsto \color{blue}{100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot n\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 78.2%

      \[\leadsto \color{blue}{100 \cdot n} \]
   6. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto \color{blue}{n \cdot 100} \]

   7. Simplified 78.2%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -122000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 28000000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 19.0% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 26.7%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Step-by-step derivation

   1. associate-*r/ 26.7%

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

   2. sub-neg 26.7%

      \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

   3. distribute-lft-in 26.7%

      \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

   4. metadata-eval 26.7%

      \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

   5. metadata-eval 26.7%

      \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

   6. metadata-eval 26.7%

      \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

   7. fma-define 26.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

   8. metadata-eval 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

3. Simplified 26.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 26.7%

      \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

   2. *-commutative 26.7%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

6. Applied egg-rr 26.7%

   \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

7. Taylor expanded in i around 0 21.8%

   \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

8. Taylor expanded in i around 0 22.0%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 22.0%

   \[\leadsto 0 \]
10. Add Preprocessing

Developer target: 33.0% 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 2024036 
(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))))