Compound Interest

Percentage Accurate: 27.3% → 98.9%

Time: 19.7s

Alternatives: 18

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1000:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{fma}\left(100, t_0, -100\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1000.0)
     (* 100.0 (+ (* (/ n i) 0.0) (- (* n (/ t_0 i)) (/ n i))))
     (if (<= t_1 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (/ n (/ i (fma 100.0 t_0 -100.0)))
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334)))))))))

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 <= -1000.0) {
		tmp = 100.0 * (((n / i) * 0.0) + ((n * (t_0 / i)) - (n / i)));
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n / (i / fma(100.0, t_0, -100.0));
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	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 <= -1000.0)
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) * 0.0) + Float64(Float64(n * Float64(t_0 / i)) - Float64(n / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n / Float64(i / fma(100.0, t_0, -100.0)));
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	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, -1000.0], N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] * 0.0), $MachinePrecision] + N[(N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n / N[(i / N[(100.0 * t$95$0 + -100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $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)) < -1e3

   1. Initial program 99.7%

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

      1. div-sub 99.7%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. prod-diff 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-udef 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-lft-out 100.0%

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

      7. metadata-eval 100.0%

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

      8. fma-udef 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -1e3 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 28.7%

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

      1. associate-/r/ 28.6%

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

      2. associate-*r* 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-*r/ 28.6%

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

      5. sub-neg 28.6%

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

      6. distribute-lft-in 28.6%

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

      7. fma-def 28.6%

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

      8. metadata-eval 28.6%

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

      9. metadata-eval 28.6%

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

   3. Simplified 28.6%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 24.9%

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

   5. Applied egg-rr 72.5%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. associate-*r/ 98.7%

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

      5. associate-*l* 98.7%

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

      6. associate-*r/ 98.8%

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

      7. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

      1. associate-*r/ 98.8%

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

   9. Applied egg-rr 98.8%

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

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

   1. Initial program 99.7%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. fma-def 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-udef 1.9%

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

      2. *-commutative 1.9%

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

   5. Applied egg-rr 1.9%

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

   6. Taylor expanded in n around inf 1.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-def 81.0%

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

   8. Simplified 81.0%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.0%

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

      3. associate-/r* 81.0%

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

      4. div-inv 81.1%

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

      5. metadata-eval 81.1%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 99.9%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1000:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t_0 \cdot 100\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))) (t_1 (* t_0 100.0)))
   (if (<= t_0 -5e-253)
     t_1
     (if (<= t_0 0.0)
       (* n (* 100.0 (/ (expm1 i) i)))
       (if (<= t_0 INFINITY)
         t_1
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334)))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -5e-253) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	t_1 = Float64(t_0 * 100.0)
	tmp = 0.0
	if (t_0 <= -5e-253)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-253], t$95$1, If[LessEqual[t$95$0, 0.0], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.99999999999999971e-253 or 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.9%

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

   if -4.99999999999999971e-253 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 25.9%

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

   2. Taylor expanded in n around inf 37.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. expm1-def 73.2%

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

   4. Simplified 73.2%

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

      1. clear-num 73.1%

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

      2. un-div-inv 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. associate-/l* 73.1%

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

      2. associate-/r/ 74.1%

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

      3. associate-*r/ 74.2%

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

   8. Simplified 74.2%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-udef 1.9%

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

      2. *-commutative 1.9%

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

   5. Applied egg-rr 1.9%

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

   6. Taylor expanded in n around inf 1.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-def 81.0%

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

   8. Simplified 81.0%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.0%

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

      3. associate-/r* 81.0%

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

      4. div-inv 81.1%

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

      5. metadata-eval 81.1%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 99.9%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 3: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-253)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* n (* 100.0 (/ (expm1 i) i)))
       (if (<= t_1 INFINITY)
         (* n (/ (+ -100.0 (* t_0 100.0)) i))
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334)))))))))

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 <= -5e-253) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	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 <= -5e-253)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / i));
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	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, -5e-253], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $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)) < -4.99999999999999971e-253

   1. Initial program 97.9%

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

   if -4.99999999999999971e-253 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 25.9%

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

   2. Taylor expanded in n around inf 37.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. expm1-def 73.2%

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

   4. Simplified 73.2%

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

      1. clear-num 73.1%

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

      2. un-div-inv 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. associate-/l* 73.1%

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

      2. associate-/r/ 74.1%

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

      3. associate-*r/ 74.2%

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

   8. Simplified 74.2%

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

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

   1. Initial program 99.7%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. fma-def 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-udef 1.9%

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

      2. *-commutative 1.9%

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

   5. Applied egg-rr 1.9%

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

   6. Taylor expanded in n around inf 1.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-def 81.0%

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

   8. Simplified 81.0%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.0%

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

      3. associate-/r* 81.0%

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

      4. div-inv 81.1%

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

      5. metadata-eval 81.1%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 99.9%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 4: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;n \cdot \frac{1 - t_0}{i \cdot -0.01}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-253)
     (* n (/ (- 1.0 t_0) (* i -0.01)))
     (if (<= t_1 0.0)
       (* n (* 100.0 (/ (expm1 i) i)))
       (if (<= t_1 INFINITY)
         (* n (/ (+ -100.0 (* t_0 100.0)) i))
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334)))))))))

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 <= -5e-253) {
		tmp = n * ((1.0 - t_0) / (i * -0.01));
	} else if (t_1 <= 0.0) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	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 <= -5e-253)
		tmp = Float64(n * Float64(Float64(1.0 - t_0) / Float64(i * -0.01)));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / i));
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	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, -5e-253], N[(n * N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(i * -0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $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)) < -4.99999999999999971e-253

   1. Initial program 97.9%

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

      1. associate-/r/ 97.9%

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

      2. associate-*r* 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-*r/ 97.8%

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

      5. sub-neg 97.8%

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

      6. distribute-lft-in 97.7%

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

      7. fma-def 97.8%

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

      8. metadata-eval 97.8%

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

      9. metadata-eval 97.8%

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

   3. Simplified 97.8%

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

      1. expm1-log1p-u 54.0%

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

      2. expm1-udef 6.1%

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

   5. Applied egg-rr 6.1%

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

      1. expm1-def 55.8%

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

      2. expm1-log1p 55.8%

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

      3. *-rgt-identity 55.8%

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

      4. associate-*r/ 55.6%

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

      5. associate-*l* 55.6%

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

      6. associate-*r/ 55.7%

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

      7. metadata-eval 55.7%

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

   7. Simplified 55.7%

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

      1. associate-*r/ 55.8%

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

   9. Applied egg-rr 55.8%

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

       1. associate-/l* 55.8%

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

       2. expm1-udef 54.1%

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

       3. div-sub 54.1%

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

       4. *-commutative 54.1%

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

       5. log1p-udef 54.1%

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

       6. exp-to-pow 97.9%

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

       7. +-commutative 97.9%

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

       8. div-inv 97.9%

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

       9. metadata-eval 97.9%

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

       10. div-inv 97.9%

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

       11. metadata-eval 97.9%

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

   11. Applied egg-rr 97.9%

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

       1. frac-2neg 97.9%

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

       2. frac-2neg 97.9%

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

       3. metadata-eval 97.9%

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

       4. sub-div 97.9%

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

   13. Applied egg-rr 97.9%

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

       1. sub-neg 97.9%

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

       2. +-commutative 97.9%

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

       3. metadata-eval 97.9%

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

       4. distribute-rgt-neg-in 97.9%

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

       5. metadata-eval 97.9%

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

   15. Simplified 97.9%

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

   if -4.99999999999999971e-253 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 25.9%

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

   2. Taylor expanded in n around inf 37.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. expm1-def 73.2%

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

   4. Simplified 73.2%

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

      1. clear-num 73.1%

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

      2. un-div-inv 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. associate-/l* 73.1%

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

      2. associate-/r/ 74.1%

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

      3. associate-*r/ 74.2%

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

   8. Simplified 74.2%

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

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

   1. Initial program 99.7%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. fma-def 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-udef 1.9%

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

      2. *-commutative 1.9%

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

   5. Applied egg-rr 1.9%

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

   6. Taylor expanded in n around inf 1.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-def 81.0%

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

   8. Simplified 81.0%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.0%

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

      3. associate-/r* 81.0%

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

      4. div-inv 81.1%

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

      5. metadata-eval 81.1%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 99.9%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-253}:\\ \;\;\;\;n \cdot \frac{1 - {\left(1 + \frac{i}{n}\right)}^{n}}{i \cdot -0.01}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 5: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;n \cdot \frac{\frac{t_0}{i} \cdot 0.01 + 0.01 \cdot \frac{-1}{i}}{0.0001}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-253)
     (* n (/ (+ (* (/ t_0 i) 0.01) (* 0.01 (/ -1.0 i))) 0.0001))
     (if (<= t_1 0.0)
       (* n (* 100.0 (/ (expm1 i) i)))
       (if (<= t_1 INFINITY)
         (* n (/ (+ -100.0 (* t_0 100.0)) i))
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334)))))))))

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 <= -5e-253) {
		tmp = n * ((((t_0 / i) * 0.01) + (0.01 * (-1.0 / i))) / 0.0001);
	} else if (t_1 <= 0.0) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	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 <= -5e-253)
		tmp = Float64(n * Float64(Float64(Float64(Float64(t_0 / i) * 0.01) + Float64(0.01 * Float64(-1.0 / i))) / 0.0001));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / i));
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	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, -5e-253], N[(n * N[(N[(N[(N[(t$95$0 / i), $MachinePrecision] * 0.01), $MachinePrecision] + N[(0.01 * N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.0001), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $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)) < -4.99999999999999971e-253

   1. Initial program 97.9%

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

      1. associate-/r/ 97.9%

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

      2. associate-*r* 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-*r/ 97.8%

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

      5. sub-neg 97.8%

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

      6. distribute-lft-in 97.7%

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

      7. fma-def 97.8%

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

      8. metadata-eval 97.8%

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

      9. metadata-eval 97.8%

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

   3. Simplified 97.8%

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

      1. expm1-log1p-u 54.0%

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

      2. expm1-udef 6.1%

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

   5. Applied egg-rr 6.1%

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

      1. expm1-def 55.8%

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

      2. expm1-log1p 55.8%

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

      3. *-rgt-identity 55.8%

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

      4. associate-*r/ 55.6%

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

      5. associate-*l* 55.6%

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

      6. associate-*r/ 55.7%

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

      7. metadata-eval 55.7%

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

   7. Simplified 55.7%

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

      1. associate-*r/ 55.8%

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

   9. Applied egg-rr 55.8%

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

       1. associate-/l* 55.8%

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

       2. expm1-udef 54.1%

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

       3. div-sub 54.1%

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

       4. *-commutative 54.1%

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

       5. log1p-udef 54.1%

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

       6. exp-to-pow 97.9%

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

       7. +-commutative 97.9%

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

       8. div-inv 97.9%

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

       9. metadata-eval 97.9%

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

       10. div-inv 97.9%

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

       11. metadata-eval 97.9%

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

   11. Applied egg-rr 97.9%

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

       1. associate-/r* 97.9%

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

       2. associate-/r* 97.9%

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

       3. frac-sub 98.1%

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

       4. metadata-eval 98.1%

          \[\leadsto n \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot 0.01 - 0.01 \cdot \frac{1}{i}}{\color{blue}{0.0001}} \]

   13. Applied egg-rr 98.1%

       \[\leadsto n \cdot \color{blue}{\frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot 0.01 - 0.01 \cdot \frac{1}{i}}{0.0001}} \]

   if -4.99999999999999971e-253 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 25.9%

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

   2. Taylor expanded in n around inf 37.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. expm1-def 73.2%

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

   4. Simplified 73.2%

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

      1. clear-num 73.1%

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

      2. un-div-inv 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. associate-/l* 73.1%

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

      2. associate-/r/ 74.1%

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

      3. associate-*r/ 74.2%

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

   8. Simplified 74.2%

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

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

   1. Initial program 99.7%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. fma-def 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-udef 1.9%

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

      2. *-commutative 1.9%

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

   5. Applied egg-rr 1.9%

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

   6. Taylor expanded in n around inf 1.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-def 81.0%

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

   8. Simplified 81.0%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.0%

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

      3. associate-/r* 81.0%

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

      4. div-inv 81.1%

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

      5. metadata-eval 81.1%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 99.9%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-253}:\\ \;\;\;\;n \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot 0.01 + 0.01 \cdot \frac{-1}{i}}{0.0001}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 6: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1000:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1000.0)
     (* 100.0 (+ (* (/ n i) 0.0) (- (* n (/ t_0 i)) (/ n i))))
     (if (<= t_1 0.0)
       (* n (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 i)))
       (if (<= t_1 INFINITY)
         (* n (/ (+ -100.0 (* t_0 100.0)) i))
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334)))))))))

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 <= -1000.0) {
		tmp = 100.0 * (((n / i) * 0.0) + ((n * (t_0 / i)) - (n / i)));
	} else if (t_1 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	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 <= -1000.0)
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) * 0.0) + Float64(Float64(n * Float64(t_0 / i)) - Float64(n / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / i));
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	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, -1000.0], N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] * 0.0), $MachinePrecision] + N[(N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $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)) < -1e3

   1. Initial program 99.7%

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

      1. div-sub 99.7%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. prod-diff 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-udef 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-lft-out 100.0%

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

      7. metadata-eval 100.0%

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

      8. fma-udef 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -1e3 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 28.7%

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

      1. associate-/r/ 28.6%

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

      2. associate-*r* 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-*r/ 28.6%

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

      5. sub-neg 28.6%

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

      6. distribute-lft-in 28.6%

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

      7. fma-def 28.6%

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

      8. metadata-eval 28.6%

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

      9. metadata-eval 28.6%

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

   3. Simplified 28.6%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 24.9%

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

   5. Applied egg-rr 72.5%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. associate-*r/ 98.7%

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

      5. associate-*l* 98.7%

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

      6. associate-*r/ 98.8%

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

      7. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

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

   1. Initial program 99.7%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. fma-def 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-udef 1.9%

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

      2. *-commutative 1.9%

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

   5. Applied egg-rr 1.9%

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

   6. Taylor expanded in n around inf 1.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-def 81.0%

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

   8. Simplified 81.0%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.0%

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

      3. associate-/r* 81.0%

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

      4. div-inv 81.1%

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

      5. metadata-eval 81.1%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 99.9%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1000:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 7: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1000:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1000.0)
     (* 100.0 (+ (* (/ n i) 0.0) (- (* n (/ t_0 i)) (/ n i))))
     (if (<= t_1 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* n (/ (+ -100.0 (* t_0 100.0)) i))
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334)))))))))

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 <= -1000.0) {
		tmp = 100.0 * (((n / i) * 0.0) + ((n * (t_0 / i)) - (n / i)));
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	}
	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 <= -1000.0)
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) * 0.0) + Float64(Float64(n * Float64(t_0 / i)) - Float64(n / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / i));
	else
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	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, -1000.0], N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] * 0.0), $MachinePrecision] + N[(N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $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)) < -1e3

   1. Initial program 99.7%

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

      1. div-sub 99.7%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. prod-diff 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-udef 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-lft-out 100.0%

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

      7. metadata-eval 100.0%

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

      8. fma-udef 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -1e3 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 28.7%

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

      1. associate-/r/ 28.6%

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

      2. associate-*r* 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-*r/ 28.6%

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

      5. sub-neg 28.6%

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

      6. distribute-lft-in 28.6%

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

      7. fma-def 28.6%

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

      8. metadata-eval 28.6%

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

      9. metadata-eval 28.6%

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

   3. Simplified 28.6%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 24.9%

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

   5. Applied egg-rr 72.5%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. associate-*r/ 98.7%

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

      5. associate-*l* 98.7%

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

      6. associate-*r/ 98.8%

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

      7. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

      1. associate-*r/ 98.8%

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

   9. Applied egg-rr 98.8%

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

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

   1. Initial program 99.7%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. fma-def 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-udef 1.9%

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

      2. *-commutative 1.9%

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

   5. Applied egg-rr 1.9%

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

   6. Taylor expanded in n around inf 1.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-def 81.0%

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

   8. Simplified 81.0%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.0%

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

      3. associate-/r* 81.0%

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

      4. div-inv 81.1%

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

      5. metadata-eval 81.1%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 99.9%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 99.9%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 99.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1000:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot 0 + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \end{array} \]

Alternative 8: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.6 \cdot 10^{-236}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.28 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+52} \lor \neg \left(n \leq 5 \cdot 10^{+109}\right) \land n \leq 5.1 \cdot 10^{+117}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -1.6e-236)
     t_0
     (if (<= n 1.28e-197)
       (* 100.0 (/ 0.0 (/ i n)))
       (if (<= n 4.8e-18)
         (/ n (+ 0.01 (fma -0.005 i (* (* i i) 0.0008333333333333334))))
         (if (or (<= n 4e+52) (and (not (<= n 5e+109)) (<= n 5.1e+117)))
           (* 100.0 (/ (+ i (* (* i i) (- 0.5 (/ 0.5 n)))) (/ i n)))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -1.6e-236) {
		tmp = t_0;
	} else if (n <= 1.28e-197) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 4.8e-18) {
		tmp = n / (0.01 + fma(-0.005, i, ((i * i) * 0.0008333333333333334)));
	} else if ((n <= 4e+52) || (!(n <= 5e+109) && (n <= 5.1e+117))) {
		tmp = 100.0 * ((i + ((i * i) * (0.5 - (0.5 / n)))) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.6e-236)
		tmp = t_0;
	elseif (n <= 1.28e-197)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	elseif (n <= 4.8e-18)
		tmp = Float64(n / Float64(0.01 + fma(-0.005, i, Float64(Float64(i * i) * 0.0008333333333333334))));
	elseif ((n <= 4e+52) || (!(n <= 5e+109) && (n <= 5.1e+117)))
		tmp = Float64(100.0 * Float64(Float64(i + Float64(Float64(i * i) * Float64(0.5 - Float64(0.5 / n)))) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.6e-236], t$95$0, If[LessEqual[n, 1.28e-197], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.8e-18], N[(n / N[(0.01 + N[(-0.005 * i + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[n, 4e+52], And[N[Not[LessEqual[n, 5e+109]], $MachinePrecision], LessEqual[n, 5.1e+117]]], N[(100.0 * N[(N[(i + N[(N[(i * i), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.6e-236 or 4e52 < n < 5.0000000000000001e109 or 5.0999999999999996e117 < n

   1. Initial program 24.7%

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

   2. Taylor expanded in n around inf 33.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. expm1-def 62.5%

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

   4. Simplified 62.5%

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

      1. clear-num 62.4%

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

      2. un-div-inv 62.4%

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

   6. Applied egg-rr 62.4%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 83.1%

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

      3. associate-*r/ 83.2%

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

   8. Simplified 83.2%

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

   if -1.6e-236 < n < 1.27999999999999998e-197

   1. Initial program 65.1%

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

   2. Taylor expanded in i around 0 83.5%

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

   if 1.27999999999999998e-197 < n < 4.79999999999999988e-18

   1. Initial program 19.5%

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

      1. associate-/r/ 19.6%

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

      2. associate-*r* 19.6%

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

      3. *-commutative 19.6%

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

      4. associate-*r/ 19.6%

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

      5. sub-neg 19.6%

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

      6. distribute-lft-in 19.6%

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

      7. fma-def 19.6%

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

      8. metadata-eval 19.6%

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

      9. metadata-eval 19.6%

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

   3. Simplified 19.6%

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

      1. fma-udef 19.6%

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

      2. *-commutative 19.6%

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

   5. Applied egg-rr 19.6%

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

   6. Taylor expanded in n around inf 3.8%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 3.8%

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

      2. metadata-eval 3.8%

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

      3. metadata-eval 3.8%

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

      4. distribute-lft-in 3.8%

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

      5. metadata-eval 3.8%

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

      6. sub-neg 3.8%

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

      7. expm1-def 51.5%

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

   8. Simplified 51.5%

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

      1. clear-num 51.4%

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

      2. un-div-inv 51.3%

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

      3. associate-/r* 51.4%

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

      4. div-inv 51.4%

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

      5. metadata-eval 51.4%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 51.4%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 66.7%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   12. Step-by-step derivation

       1. fma-def 66.7%

          \[\leadsto \frac{n}{0.01 + \color{blue}{\mathsf{fma}\left(-0.005, i, 0.0008333333333333334 \cdot {i}^{2}\right)}} \]

       2. *-commutative 66.7%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{{i}^{2} \cdot 0.0008333333333333334}\right)} \]

       3. unpow2 66.7%

          \[\leadsto \frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334\right)} \]

   13. Simplified 66.7%

       \[\leadsto \frac{n}{\color{blue}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}} \]

   if 4.79999999999999988e-18 < n < 4e52 or 5.0000000000000001e109 < n < 5.0999999999999996e117

   1. Initial program 62.7%

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

   2. Taylor expanded in i around 0 92.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. unpow2 92.7%

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

      2. associate-*r/ 92.7%

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

      3. metadata-eval 92.7%

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

   4. Simplified 92.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-236}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.28 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{n}{0.01 + \mathsf{fma}\left(-0.005, i, \left(i \cdot i\right) \cdot 0.0008333333333333334\right)}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+52} \lor \neg \left(n \leq 5 \cdot 10^{+109}\right) \land n \leq 5.1 \cdot 10^{+117}:\\ \;\;\;\;100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 9: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ t_1 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-236}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.28 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-169}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 i)) i))) (t_1 (* 100.0 (/ 0.0 (/ i n)))))
   (if (<= n -1.25e-236)
     t_0
     (if (<= n 1.28e-197)
       t_1
       (if (<= n 2.9e-169)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.1e-141) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1(i)) / i);
	double t_1 = 100.0 * (0.0 / (i / n));
	double tmp;
	if (n <= -1.25e-236) {
		tmp = t_0;
	} else if (n <= 1.28e-197) {
		tmp = t_1;
	} else if (n <= 2.9e-169) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.1e-141) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 * Math.expm1(i)) / i);
	double t_1 = 100.0 * (0.0 / (i / n));
	double tmp;
	if (n <= -1.25e-236) {
		tmp = t_0;
	} else if (n <= 1.28e-197) {
		tmp = t_1;
	} else if (n <= 2.9e-169) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.1e-141) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 * math.expm1(i)) / i)
	t_1 = 100.0 * (0.0 / (i / n))
	tmp = 0
	if n <= -1.25e-236:
		tmp = t_0
	elif n <= 1.28e-197:
		tmp = t_1
	elif n <= 2.9e-169:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.1e-141:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
	t_1 = Float64(100.0 * Float64(0.0 / Float64(i / n)))
	tmp = 0.0
	if (n <= -1.25e-236)
		tmp = t_0;
	elseif (n <= 1.28e-197)
		tmp = t_1;
	elseif (n <= 2.9e-169)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.1e-141)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.25e-236], t$95$0, If[LessEqual[n, 1.28e-197], t$95$1, If[LessEqual[n, 2.9e-169], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.1e-141], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2499999999999999e-236 or 1.10000000000000005e-141 < n

   1. Initial program 25.9%

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

      1. associate-/r/ 26.3%

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

      2. associate-*r* 26.2%

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

      3. *-commutative 26.2%

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

      4. associate-*r/ 26.3%

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

      5. sub-neg 26.3%

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

      6. distribute-lft-in 26.3%

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

      7. fma-def 26.3%

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

      8. metadata-eval 26.3%

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

      9. metadata-eval 26.3%

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

   3. Simplified 26.3%

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

      1. fma-udef 26.3%

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

      2. *-commutative 26.3%

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

   5. Applied egg-rr 26.3%

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

   6. Taylor expanded in n around inf 31.1%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 31.1%

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

      2. metadata-eval 31.1%

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

      3. metadata-eval 31.1%

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

      4. distribute-lft-in 31.1%

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

      5. metadata-eval 31.1%

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

      6. sub-neg 31.1%

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

      7. expm1-def 80.3%

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

   8. Simplified 80.3%

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

   if -1.2499999999999999e-236 < n < 1.27999999999999998e-197 or 2.90000000000000019e-169 < n < 1.10000000000000005e-141

   1. Initial program 61.8%

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

   2. Taylor expanded in i around 0 78.5%

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

   if 1.27999999999999998e-197 < n < 2.90000000000000019e-169

   1. Initial program 5.6%

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

   2. Taylor expanded in i around 0 76.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-236}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.28 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-169}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]

Alternative 10: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ t_1 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1.8 \cdot 10^{-235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-169}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))) (t_1 (* 100.0 (/ 0.0 (/ i n)))))
   (if (<= n -1.8e-235)
     t_0
     (if (<= n 1.35e-197)
       t_1
       (if (<= n 2.9e-169)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.6e-141) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double t_1 = 100.0 * (0.0 / (i / n));
	double tmp;
	if (n <= -1.8e-235) {
		tmp = t_0;
	} else if (n <= 1.35e-197) {
		tmp = t_1;
	} else if (n <= 2.9e-169) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.6e-141) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double t_1 = 100.0 * (0.0 / (i / n));
	double tmp;
	if (n <= -1.8e-235) {
		tmp = t_0;
	} else if (n <= 1.35e-197) {
		tmp = t_1;
	} else if (n <= 2.9e-169) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.6e-141) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	t_1 = 100.0 * (0.0 / (i / n))
	tmp = 0
	if n <= -1.8e-235:
		tmp = t_0
	elif n <= 1.35e-197:
		tmp = t_1
	elif n <= 2.9e-169:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.6e-141:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	t_1 = Float64(100.0 * Float64(0.0 / Float64(i / n)))
	tmp = 0.0
	if (n <= -1.8e-235)
		tmp = t_0;
	elseif (n <= 1.35e-197)
		tmp = t_1;
	elseif (n <= 2.9e-169)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.6e-141)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.8e-235], t$95$0, If[LessEqual[n, 1.35e-197], t$95$1, If[LessEqual[n, 2.9e-169], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e-141], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.79999999999999999e-235 or 1.6000000000000001e-141 < n

   1. Initial program 25.9%

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

   2. Taylor expanded in n around inf 30.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. expm1-def 62.3%

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

   4. Simplified 62.3%

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

      1. clear-num 62.2%

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

      2. un-div-inv 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. associate-/l* 62.2%

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

      2. associate-/r/ 80.3%

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

      3. associate-*r/ 80.4%

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

   8. Simplified 80.4%

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

   if -1.79999999999999999e-235 < n < 1.35000000000000009e-197 or 2.90000000000000019e-169 < n < 1.6000000000000001e-141

   1. Initial program 61.8%

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

   2. Taylor expanded in i around 0 78.5%

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

   if 1.35000000000000009e-197 < n < 2.90000000000000019e-169

   1. Initial program 5.6%

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

   2. Taylor expanded in i around 0 76.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{-235}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-169}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 11: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{-25} \lor \neg \left(i \leq 0.0009\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2e-25) (not (<= i 0.0009)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2e-25) || !(i <= 0.0009)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2e-25) || !(i <= 0.0009)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2e-25) or not (i <= 0.0009):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2e-25) || !(i <= 0.0009))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2e-25], N[Not[LessEqual[i, 0.0009]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2.00000000000000008e-25 or 8.9999999999999998e-4 < i

   1. Initial program 55.9%

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

   2. Taylor expanded in n around inf 54.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. expm1-def 53.8%

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

   4. Simplified 53.8%

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

   if -2.00000000000000008e-25 < i < 8.9999999999999998e-4

   1. Initial program 9.5%

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

   2. Taylor expanded in i around 0 86.1%

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

      1. *-commutative 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{-25} \lor \neg \left(i \leq 0.0009\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 12: 64.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-234}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-169}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ 0.0 (/ i n)))))
   (if (<= n -6.2e-234)
     (/ n (+ 0.01 (* i -0.005)))
     (if (<= n 2e-197)
       t_0
       (if (<= n 2.9e-169)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.1e-141) t_0 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (0.0 / (i / n));
	double tmp;
	if (n <= -6.2e-234) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 2e-197) {
		tmp = t_0;
	} else if (n <= 2.9e-169) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.1e-141) {
		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 = 100.0d0 * (0.0d0 / (i / n))
    if (n <= (-6.2d-234)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 2d-197) then
        tmp = t_0
    else if (n <= 2.9d-169) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 1.1d-141) 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 = 100.0 * (0.0 / (i / n));
	double tmp;
	if (n <= -6.2e-234) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 2e-197) {
		tmp = t_0;
	} else if (n <= 2.9e-169) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.1e-141) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (0.0 / (i / n))
	tmp = 0
	if n <= -6.2e-234:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 2e-197:
		tmp = t_0
	elif n <= 2.9e-169:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.1e-141:
		tmp = t_0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(0.0 / Float64(i / n)))
	tmp = 0.0
	if (n <= -6.2e-234)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 2e-197)
		tmp = t_0;
	elseif (n <= 2.9e-169)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.1e-141)
		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 = 100.0 * (0.0 / (i / n));
	tmp = 0.0;
	if (n <= -6.2e-234)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 2e-197)
		tmp = t_0;
	elseif (n <= 2.9e-169)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 1.1e-141)
		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[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-234], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2e-197], t$95$0, If[LessEqual[n, 2.9e-169], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.1e-141], 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 < -6.2000000000000003e-234

   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/ 28.8%

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

      2. associate-*r* 28.8%

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

      3. *-commutative 28.8%

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

      4. associate-*r/ 28.9%

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

      5. sub-neg 28.9%

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

      6. distribute-lft-in 28.9%

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

      7. fma-def 28.9%

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

      8. metadata-eval 28.9%

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

      9. metadata-eval 28.9%

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

   3. Simplified 28.9%

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

      1. fma-udef 28.9%

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

      2. *-commutative 28.9%

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

   5. Applied egg-rr 28.9%

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

   6. Taylor expanded in n around inf 32.9%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 32.9%

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

      2. metadata-eval 32.9%

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

      3. metadata-eval 32.9%

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

      4. distribute-lft-in 32.9%

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

      5. metadata-eval 32.9%

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

      6. sub-neg 32.9%

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

      7. expm1-def 76.5%

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

   8. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.5%

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

      3. associate-/r* 76.5%

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

      4. div-inv 76.5%

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

      5. metadata-eval 76.5%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 76.5%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 56.5%

       \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   12. Step-by-step derivation

       1. *-commutative 56.5%

          \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   13. Simplified 56.5%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -6.2000000000000003e-234 < n < 2e-197 or 2.90000000000000019e-169 < n < 1.10000000000000005e-141

   1. Initial program 61.8%

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

   2. Taylor expanded in i around 0 78.5%

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

   if 2e-197 < n < 2.90000000000000019e-169

   1. Initial program 5.6%

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

   2. Taylor expanded in i around 0 76.4%

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

   if 1.10000000000000005e-141 < n

   1. Initial program 21.8%

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

      1. associate-/r/ 22.3%

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

      2. associate-*r* 22.3%

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

      3. *-commutative 22.3%

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

      4. associate-*r/ 22.3%

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

      5. sub-neg 22.3%

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

      6. distribute-lft-in 22.3%

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

      7. fma-def 22.3%

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

      8. metadata-eval 22.3%

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

      9. metadata-eval 22.3%

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

   3. Simplified 22.3%

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

      1. fma-udef 22.3%

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

      2. *-commutative 22.3%

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

   5. Applied egg-rr 22.3%

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

   6. Taylor expanded in n around inf 28.3%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 28.3%

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

      2. metadata-eval 28.3%

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

      3. metadata-eval 28.3%

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

      4. distribute-lft-in 28.3%

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

      5. metadata-eval 28.3%

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

      6. sub-neg 28.3%

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

      7. expm1-def 86.1%

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

   8. Simplified 86.1%

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

   9. Taylor expanded in i around 0 73.2%

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

       1. *-commutative 73.2%

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

   11. Simplified 73.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-234}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-169}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13: 64.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+33} \lor \neg \left(n \leq 8 \cdot 10^{-14}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.4e+33) (not (<= n 8e-14)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e+33) || !(n <= 8e-14)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.4d+33)) .or. (.not. (n <= 8d-14))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e+33) || !(n <= 8e-14)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.4e+33) or not (n <= 8e-14):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.4e+33) || !(n <= 8e-14))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.4e+33) || ~((n <= 8e-14)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.4e+33], N[Not[LessEqual[n, 8e-14]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e33 or 7.99999999999999999e-14 < n

   1. Initial program 24.6%

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

      1. associate-/r/ 25.1%

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

      2. associate-*r* 25.1%

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

      3. *-commutative 25.1%

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

      4. associate-*r/ 25.1%

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

      5. sub-neg 25.1%

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

      6. distribute-lft-in 25.1%

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

      7. fma-def 25.1%

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

      8. metadata-eval 25.1%

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

      9. metadata-eval 25.1%

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

   3. Simplified 25.1%

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

      1. fma-udef 25.1%

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

      2. *-commutative 25.1%

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

   5. Applied egg-rr 25.1%

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

   6. Taylor expanded in n around inf 39.0%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 39.0%

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

      2. metadata-eval 39.0%

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

      3. metadata-eval 39.0%

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

      4. distribute-lft-in 39.0%

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

      5. metadata-eval 39.0%

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

      6. sub-neg 39.0%

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

      7. expm1-def 87.4%

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

   8. Simplified 87.4%

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

   9. Taylor expanded in i around 0 61.1%

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

       1. *-commutative 61.1%

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

   11. Simplified 61.1%

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

   if -1.4e33 < n < 7.99999999999999999e-14

   1. Initial program 40.4%

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

   2. Taylor expanded in i around 0 57.8%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+33} \lor \neg \left(n \leq 8 \cdot 10^{-14}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 14: 59.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+57}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -5e+57)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 1.2e+64) (* n 100.0) (* 50.0 (* i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -5e+57) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.2e+64) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-5d+57)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 1.2d+64) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -5e+57) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.2e+64) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -5e+57:
		tmp = 100.0 * (i / (i / n))
	elif i <= 1.2e+64:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -5e+57)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 1.2e+64)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -5e+57)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 1.2e+64)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -5e+57], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.2e+64], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.99999999999999972e57

   1. Initial program 61.3%

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

   2. Taylor expanded in i around 0 26.2%

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

   if -4.99999999999999972e57 < i < 1.2e64

   1. Initial program 16.6%

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

   2. Taylor expanded in i around 0 71.8%

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

      1. *-commutative 71.8%

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

   4. Simplified 71.8%

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

   if 1.2e64 < i

   1. Initial program 55.1%

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

      1. associate-/r/ 55.4%

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

      2. associate-*r* 55.3%

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

      3. *-commutative 55.3%

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

      4. associate-*r/ 55.5%

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

      5. sub-neg 55.5%

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

      6. distribute-lft-in 55.5%

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

      7. fma-def 55.5%

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

      8. metadata-eval 55.5%

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

      9. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

      1. fma-udef 55.5%

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

      2. *-commutative 55.5%

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

   5. Applied egg-rr 55.5%

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

   6. Taylor expanded in n around inf 41.2%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 41.2%

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

      2. metadata-eval 41.2%

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

      3. metadata-eval 41.2%

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

      4. distribute-lft-in 41.2%

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

      5. metadata-eval 41.2%

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

      6. sub-neg 41.2%

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

      7. expm1-def 41.2%

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

   8. Simplified 41.2%

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

   9. Taylor expanded in i around 0 28.6%

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

   10. Taylor expanded in i around inf 28.6%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+57}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 15: 63.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.25 \cdot 10^{-130}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.25e-130) (/ n (+ 0.01 (* i -0.005))) (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.25e-130) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 2.25d-130) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 2.25e-130) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 2.25e-130:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.25e-130)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 2.25e-130)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.25e-130], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.25e-130

   1. Initial program 36.6%

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

      1. associate-/r/ 36.8%

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

      2. associate-*r* 36.8%

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

      3. *-commutative 36.8%

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

      4. associate-*r/ 36.8%

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

      5. sub-neg 36.8%

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

      6. distribute-lft-in 36.8%

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

      7. fma-def 36.8%

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

      8. metadata-eval 36.8%

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

      9. metadata-eval 36.8%

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

   3. Simplified 36.8%

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

      1. fma-udef 36.8%

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

      2. *-commutative 36.8%

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

   5. Applied egg-rr 36.8%

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

   6. Taylor expanded in n around inf 33.1%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 33.1%

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

      2. metadata-eval 33.1%

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

      3. metadata-eval 33.1%

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

      4. distribute-lft-in 33.1%

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

      5. metadata-eval 33.1%

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

      6. sub-neg 33.1%

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

      7. expm1-def 63.5%

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

   8. Simplified 63.5%

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

      1. clear-num 63.5%

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

      2. un-div-inv 63.4%

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

      3. associate-/r* 63.5%

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

      4. div-inv 63.5%

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

      5. metadata-eval 63.5%

         \[\leadsto \frac{n}{\frac{i \cdot \color{blue}{0.01}}{\mathsf{expm1}\left(i\right)}} \]

   10. Applied egg-rr 63.5%

       \[\leadsto \color{blue}{\frac{n}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(i\right)}}} \]

   11. Taylor expanded in i around 0 54.8%

       \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   12. Step-by-step derivation

       1. *-commutative 54.8%

          \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   13. Simplified 54.8%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if 2.25e-130 < n

   1. Initial program 21.8%

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

      1. associate-/r/ 22.3%

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

      2. associate-*r* 22.3%

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

      3. *-commutative 22.3%

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

      4. associate-*r/ 22.3%

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

      5. sub-neg 22.3%

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

      6. distribute-lft-in 22.3%

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

      7. fma-def 22.3%

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

      8. metadata-eval 22.3%

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

      9. metadata-eval 22.3%

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

   3. Simplified 22.3%

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

      1. fma-udef 22.3%

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

      2. *-commutative 22.3%

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

   5. Applied egg-rr 22.3%

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

   6. Taylor expanded in n around inf 28.3%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 28.3%

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

      2. metadata-eval 28.3%

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

      3. metadata-eval 28.3%

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

      4. distribute-lft-in 28.3%

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

      5. metadata-eval 28.3%

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

      6. sub-neg 28.3%

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

      7. expm1-def 86.1%

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

   8. Simplified 86.1%

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

   9. Taylor expanded in i around 0 73.2%

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

       1. *-commutative 73.2%

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

   11. Simplified 73.2%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot 50\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.25 \cdot 10^{-130}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 16: 55.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.2e+64) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.2e+64) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.2d+64) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.2e+64) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.2e+64:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.2e+64)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.2e+64)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.2e+64], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.2e64

   1. Initial program 27.5%

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

   2. Taylor expanded in i around 0 55.5%

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

      1. *-commutative 55.5%

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

   4. Simplified 55.5%

      \[\leadsto \color{blue}{n \cdot 100} \]

   if 1.2e64 < i

   1. Initial program 55.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 55.4%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 55.3%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 55.3%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 55.5%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 55.5%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 55.5%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. fma-def 55.5%

         \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{i} \]

      8. metadata-eval 55.5%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{i} \]

      9. metadata-eval 55.5%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 55.5%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
   4. Step-by-step derivation

      1. fma-udef 55.5%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{i} \]

      2. *-commutative 55.5%

         \[\leadsto n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{i} \]

   5. Applied egg-rr 55.5%

      \[\leadsto n \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{i} \]

   6. Taylor expanded in n around inf 41.2%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   7. Step-by-step derivation

      1. sub-neg 41.2%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} + \left(-100\right)}}{i} \]

      2. metadata-eval 41.2%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 41.2%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{100 \cdot -1}}{i} \]

      4. distribute-lft-in 41.2%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot \left(e^{i} + -1\right)}}{i} \]

      5. metadata-eval 41.2%

         \[\leadsto n \cdot \frac{100 \cdot \left(e^{i} + \color{blue}{\left(-1\right)}\right)}{i} \]

      6. sub-neg 41.2%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left(e^{i} - 1\right)}}{i} \]

      7. expm1-def 41.2%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   8. Simplified 41.2%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot \mathsf{expm1}\left(i\right)}}{i} \]

   9. Taylor expanded in i around 0 28.6%

      \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right) + 100 \cdot n} \]

   10. Taylor expanded in i around inf 28.6%

       \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 17: 2.8% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 32.0%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

2. Taylor expanded in i around 0 50.9%

   \[\leadsto 100 \cdot \color{blue}{\left(n + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* 51.1%

      \[\leadsto 100 \cdot \left(n + \color{blue}{\left(n \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right) \]

   2. *-commutative 51.1%

      \[\leadsto 100 \cdot \left(n + \color{blue}{\left(i \cdot n\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \]

   3. associate-*r/ 51.1%

      \[\leadsto 100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \]

   4. metadata-eval 51.1%

      \[\leadsto 100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \]

4. Simplified 51.1%

   \[\leadsto 100 \cdot \color{blue}{\left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \]

5. Taylor expanded in n around 0 2.8%

   \[\leadsto \color{blue}{-50 \cdot i} \]
6. Step-by-step derivation

   1. *-commutative 2.8%

      \[\leadsto \color{blue}{i \cdot -50} \]

7. Simplified 2.8%

   \[\leadsto \color{blue}{i \cdot -50} \]

8. Final simplification 2.8%

   \[\leadsto i \cdot -50 \]

Alternative 18: 50.4% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * 100.0d0
end function

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 32.0%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

2. Taylor expanded in i around 0 47.2%

   \[\leadsto \color{blue}{100 \cdot n} \]
3. Step-by-step derivation

   1. *-commutative 47.2%

      \[\leadsto \color{blue}{n \cdot 100} \]

4. Simplified 47.2%

   \[\leadsto \color{blue}{n \cdot 100} \]

5. Final simplification 47.2%

   \[\leadsto n \cdot 100 \]

Developer target: 33.1% 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 2023257 
(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))))