Compound Interest

Percentage Accurate: 28.5% → 97.3%

Time: 16.6s

Alternatives: 18

Speedup: 38.0×

• Could not converge on a ground truth

  1. i = 4.1204274476183495e+105
  2. n = 2.85739671831187e+231

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.5% 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: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* (/ (expm1 (* n (log1p (/ i n)))) i) (* n 100.0))
     (if (<= t_1 INFINITY)
       (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
       (/ -1.0 (- (* 0.01 (* i (/ (+ 0.5 (/ -0.5 n)) n))) (/ 0.01 n)))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) / i) * (n * 100.0)
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i) * Float64(n * 100.0));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 + Float64(-0.5 / n)) / n))) - Float64(0.01 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.8%

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

      1. associate-*r/ 24.8%

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

      2. sub-neg 24.8%

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

      3. distribute-rgt-in 24.8%

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

      4. metadata-eval 24.8%

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

      5. metadata-eval 24.8%

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

   3. Simplified 24.8%

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

      1. metadata-eval 24.8%

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

      2. metadata-eval 24.8%

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

      3. distribute-rgt-in 24.8%

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

      4. sub-neg 24.8%

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

      5. *-commutative 24.8%

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

      6. associate-*l/ 24.8%

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

      7. associate-/r/ 24.8%

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

      8. associate-*l* 24.7%

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

      9. add-exp-log 24.7%

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

      10. expm1-define 24.7%

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

      11. log-pow 32.1%

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

      12. log1p-define 97.9%

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

   6. Applied egg-rr 97.9%

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

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

   1. Initial program 98.5%

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

      1. div-sub 98.8%

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

      2. clear-num 98.4%

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

      3. sub-neg 98.4%

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

      4. div-inv 98.4%

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

      5. clear-num 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. sub-neg 98.7%

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

   6. Simplified 98.7%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. 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. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. clear-num 0.0%

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

      6. frac-2neg 0.0%

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

      7. metadata-eval 0.0%

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

      8. fma-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. distribute-neg-frac2 0.0%

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

      2. fma-define 0.0%

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

      3. +-commutative 0.0%

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

      4. distribute-neg-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

      7. distribute-lft-neg-in 0.0%

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

      8. metadata-eval 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in i around 0 99.6%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-/l* 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \color{blue}{\left(i \cdot \frac{0.5 - 0.5 \cdot \frac{1}{n}}{n}\right)} - 0.01 \cdot \frac{1}{n}} \]

       2. cancel-sign-sub-inv 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{\color{blue}{0.5 + \left(-0.5\right) \cdot \frac{1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{-0.5} \cdot \frac{1}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{\frac{-0.5 \cdot 1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{\color{blue}{-0.5}}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{\color{blue}{0.01}}{n}} \]

   11. Simplified 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-188}:\\ \;\;\;\;t\_0 \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 -5e-188)
     (* t_0 (* 100.0 (/ n i)))
     (if (<= t_1 0.0)
       (/
        -1.0
        (+
         (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n))
         (* 0.01 (/ -1.0 n))))
       (if (<= t_1 INFINITY)
         (* t_1 100.0)
         (/ -1.0 (- (* 0.01 (* i (/ (+ 0.5 (/ -0.5 n)) n))) (/ 0.01 n))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -5e-188) {
		tmp = t_0 * (100.0 * (n / i));
	} else if (t_1 <= 0.0) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1 * 100.0;
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -5e-188) {
		tmp = t_0 * (100.0 * (n / i));
	} else if (t_1 <= 0.0) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * 100.0;
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= -5e-188:
		tmp = t_0 * (100.0 * (n / i))
	elif t_1 <= 0.0:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	elif t_1 <= math.inf:
		tmp = t_1 * 100.0
	else:
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-188)
		tmp = Float64(t_0 * Float64(100.0 * Float64(n / i)));
	elseif (t_1 <= 0.0)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	elseif (t_1 <= Inf)
		tmp = Float64(t_1 * 100.0);
	else
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 + Float64(-0.5 / n)) / n))) - Float64(0.01 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = ((1.0 + (i / n)) ^ n) + -1.0;
	t_1 = t_0 / (i / n);
	tmp = 0.0;
	if (t_1 <= -5e-188)
		tmp = t_0 * (100.0 * (n / i));
	elseif (t_1 <= 0.0)
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	elseif (t_1 <= Inf)
		tmp = t_1 * 100.0;
	else
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-188], N[(t$95$0 * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$1 * 100.0), $MachinePrecision], N[(-1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -5.0000000000000001e-188

   1. Initial program 99.3%

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

      1. associate-*r/ 99.3%

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

      2. sub-neg 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. metadata-eval 99.3%

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

      2. metadata-eval 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. sub-neg 99.3%

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

      5. associate-*r/ 99.3%

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

      6. *-commutative 99.3%

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

      7. div-inv 99.3%

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

      8. clear-num 99.6%

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

      9. associate-*l* 99.6%

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

      10. add-exp-log 99.6%

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

      11. expm1-define 99.6%

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

      12. log-pow 90.5%

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

      13. log1p-define 90.5%

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

   6. Applied egg-rr 90.5%

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

      1. expm1-undefine 90.5%

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

      2. *-commutative 90.5%

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

      3. log1p-undefine 90.5%

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

      4. exp-to-pow 99.6%

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

   8. Applied egg-rr 99.6%

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

   if -5.0000000000000001e-188 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 19.8%

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

      1. associate-/r/ 19.8%

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

      2. associate-*r* 19.8%

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

      3. *-commutative 19.8%

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

      4. associate-*r/ 19.8%

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

      5. sub-neg 19.8%

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

      6. distribute-lft-in 19.8%

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

      7. metadata-eval 19.8%

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

      8. metadata-eval 19.8%

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

      9. metadata-eval 19.8%

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

      10. fma-define 19.8%

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

      11. metadata-eval 19.8%

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

   3. Simplified 19.8%

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

      1. *-commutative 19.8%

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

      2. fma-undefine 19.8%

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

      3. *-commutative 19.8%

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

      4. associate-/r/ 19.8%

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

      5. clear-num 19.8%

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

      6. frac-2neg 19.8%

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

      7. metadata-eval 19.8%

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

      8. fma-define 19.8%

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

   6. Applied egg-rr 19.8%

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

      1. distribute-neg-frac2 19.8%

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

      2. fma-define 19.8%

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

      3. +-commutative 19.8%

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

      4. distribute-neg-in 19.8%

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

      5. metadata-eval 19.8%

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

      6. *-commutative 19.8%

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

      7. distribute-lft-neg-in 19.8%

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

      8. metadata-eval 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

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

   1. Initial program 98.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. 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. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. clear-num 0.0%

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

      6. frac-2neg 0.0%

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

      7. metadata-eval 0.0%

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

      8. fma-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. distribute-neg-frac2 0.0%

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

      2. fma-define 0.0%

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

      3. +-commutative 0.0%

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

      4. distribute-neg-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

      7. distribute-lft-neg-in 0.0%

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

      8. metadata-eval 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in i around 0 99.6%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-/l* 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \color{blue}{\left(i \cdot \frac{0.5 - 0.5 \cdot \frac{1}{n}}{n}\right)} - 0.01 \cdot \frac{1}{n}} \]

       2. cancel-sign-sub-inv 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{\color{blue}{0.5 + \left(-0.5\right) \cdot \frac{1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{-0.5} \cdot \frac{1}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{\frac{-0.5 \cdot 1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{\color{blue}{-0.5}}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{\color{blue}{0.01}}{n}} \]

   11. Simplified 99.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-188}:\\ \;\;\;\;\left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-188}:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-188)
     (* n (/ (+ (* t_0 100.0) -100.0) i))
     (if (<= t_1 0.0)
       (/
        -1.0
        (+
         (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n))
         (* 0.01 (/ -1.0 n))))
       (if (<= t_1 INFINITY)
         (* t_1 100.0)
         (/ -1.0 (- (* 0.01 (* i (/ (+ 0.5 (/ -0.5 n)) n))) (/ 0.01 n))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-188) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1 * 100.0;
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-188) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * 100.0;
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -5e-188:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	elif t_1 <= 0.0:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	elif t_1 <= math.inf:
		tmp = t_1 * 100.0
	else:
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-188)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	elseif (t_1 <= Inf)
		tmp = Float64(t_1 * 100.0);
	else
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 + Float64(-0.5 / n)) / n))) - Float64(0.01 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (1.0 + (i / n)) ^ n;
	t_1 = (t_0 + -1.0) / (i / n);
	tmp = 0.0;
	if (t_1 <= -5e-188)
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	elseif (t_1 <= 0.0)
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	elseif (t_1 <= Inf)
		tmp = t_1 * 100.0;
	else
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-188], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$1 * 100.0), $MachinePrecision], N[(-1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -5.0000000000000001e-188

   1. Initial program 99.3%

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

      1. associate-/r/ 99.3%

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

      2. associate-*r* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r/ 99.4%

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

      5. sub-neg 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. metadata-eval 99.4%

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

      10. fma-define 99.4%

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

      11. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. fma-undefine 99.4%

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

      2. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   if -5.0000000000000001e-188 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 19.8%

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

      1. associate-/r/ 19.8%

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

      2. associate-*r* 19.8%

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

      3. *-commutative 19.8%

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

      4. associate-*r/ 19.8%

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

      5. sub-neg 19.8%

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

      6. distribute-lft-in 19.8%

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

      7. metadata-eval 19.8%

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

      8. metadata-eval 19.8%

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

      9. metadata-eval 19.8%

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

      10. fma-define 19.8%

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

      11. metadata-eval 19.8%

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

   3. Simplified 19.8%

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

      1. *-commutative 19.8%

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

      2. fma-undefine 19.8%

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

      3. *-commutative 19.8%

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

      4. associate-/r/ 19.8%

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

      5. clear-num 19.8%

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

      6. frac-2neg 19.8%

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

      7. metadata-eval 19.8%

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

      8. fma-define 19.8%

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

   6. Applied egg-rr 19.8%

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

      1. distribute-neg-frac2 19.8%

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

      2. fma-define 19.8%

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

      3. +-commutative 19.8%

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

      4. distribute-neg-in 19.8%

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

      5. metadata-eval 19.8%

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

      6. *-commutative 19.8%

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

      7. distribute-lft-neg-in 19.8%

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

      8. metadata-eval 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

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

   1. Initial program 98.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. 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. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. clear-num 0.0%

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

      6. frac-2neg 0.0%

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

      7. metadata-eval 0.0%

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

      8. fma-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. distribute-neg-frac2 0.0%

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

      2. fma-define 0.0%

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

      3. +-commutative 0.0%

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

      4. distribute-neg-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

      7. distribute-lft-neg-in 0.0%

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

      8. metadata-eval 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in i around 0 99.6%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-/l* 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \color{blue}{\left(i \cdot \frac{0.5 - 0.5 \cdot \frac{1}{n}}{n}\right)} - 0.01 \cdot \frac{1}{n}} \]

       2. cancel-sign-sub-inv 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{\color{blue}{0.5 + \left(-0.5\right) \cdot \frac{1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{-0.5} \cdot \frac{1}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{\frac{-0.5 \cdot 1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{\color{blue}{-0.5}}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{\color{blue}{0.01}}{n}} \]

   11. Simplified 99.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-188}:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t\_0 \cdot 100\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \end{array} \]

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-188)
     t_1
     (if (<= t_0 0.0)
       (/
        -1.0
        (+
         (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n))
         (* 0.01 (/ -1.0 n))))
       (if (<= t_0 INFINITY)
         t_1
         (/ -1.0 (- (* 0.01 (* i (/ (+ 0.5 (/ -0.5 n)) n))) (/ 0.01 n))))))))

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-188) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -5e-188) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	t_1 = t_0 * 100.0
	tmp = 0
	if t_0 <= -5e-188:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n))
	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-188)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 + Float64(-0.5 / n)) / n))) - Float64(0.01 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (((1.0 + (i / n)) ^ n) + -1.0) / (i / n);
	t_1 = t_0 * 100.0;
	tmp = 0.0;
	if (t_0 <= -5e-188)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	end
	tmp_2 = 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-188], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(-1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -5.0000000000000001e-188 or 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 98.8%

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

   if -5.0000000000000001e-188 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 19.8%

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

      1. associate-/r/ 19.8%

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

      2. associate-*r* 19.8%

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

      3. *-commutative 19.8%

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

      4. associate-*r/ 19.8%

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

      5. sub-neg 19.8%

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

      6. distribute-lft-in 19.8%

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

      7. metadata-eval 19.8%

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

      8. metadata-eval 19.8%

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

      9. metadata-eval 19.8%

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

      10. fma-define 19.8%

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

      11. metadata-eval 19.8%

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

   3. Simplified 19.8%

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

      1. *-commutative 19.8%

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

      2. fma-undefine 19.8%

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

      3. *-commutative 19.8%

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

      4. associate-/r/ 19.8%

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

      5. clear-num 19.8%

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

      6. frac-2neg 19.8%

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

      7. metadata-eval 19.8%

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

      8. fma-define 19.8%

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

   6. Applied egg-rr 19.8%

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

      1. distribute-neg-frac2 19.8%

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

      2. fma-define 19.8%

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

      3. +-commutative 19.8%

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

      4. distribute-neg-in 19.8%

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

      5. metadata-eval 19.8%

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

      6. *-commutative 19.8%

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

      7. distribute-lft-neg-in 19.8%

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

      8. metadata-eval 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. 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. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. clear-num 0.0%

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

      6. frac-2neg 0.0%

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

      7. metadata-eval 0.0%

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

      8. fma-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. distribute-neg-frac2 0.0%

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

      2. fma-define 0.0%

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

      3. +-commutative 0.0%

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

      4. distribute-neg-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

      7. distribute-lft-neg-in 0.0%

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

      8. metadata-eval 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in i around 0 99.6%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-/l* 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \color{blue}{\left(i \cdot \frac{0.5 - 0.5 \cdot \frac{1}{n}}{n}\right)} - 0.01 \cdot \frac{1}{n}} \]

       2. cancel-sign-sub-inv 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{\color{blue}{0.5 + \left(-0.5\right) \cdot \frac{1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{-0.5} \cdot \frac{1}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{\frac{-0.5 \cdot 1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{\color{blue}{-0.5}}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{\color{blue}{0.01}}{n}} \]

   11. Simplified 99.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-188}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* n (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 i)))
     (if (<= t_1 INFINITY)
       (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
       (/ -1.0 (- (* 0.01 (* i (/ (+ 0.5 (/ -0.5 n)) n))) (/ 0.01 n)))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) * (100.0 / i))
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 + Float64(-0.5 / n)) / n))) - Float64(0.01 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 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[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.8%

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

      1. associate-/r/ 24.8%

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

      2. associate-*r* 24.7%

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

      3. *-commutative 24.7%

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

      4. associate-*r/ 24.7%

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

      5. sub-neg 24.7%

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

      6. distribute-lft-in 24.7%

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

      7. metadata-eval 24.7%

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

      8. metadata-eval 24.7%

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

      9. metadata-eval 24.7%

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

      10. fma-define 24.7%

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

      11. metadata-eval 24.7%

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

   3. Simplified 24.7%

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

      1. fma-undefine 24.7%

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

      2. metadata-eval 24.7%

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

      3. metadata-eval 24.7%

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

      4. distribute-lft-in 24.7%

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

      5. sub-neg 24.7%

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

      6. *-commutative 24.7%

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

      7. add-exp-log 24.7%

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

      8. expm1-define 24.7%

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

      9. log-pow 32.1%

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

      10. log1p-define 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-/l* 97.8%

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

   8. Applied egg-rr 97.8%

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

      1. metadata-eval 97.8%

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

      2. associate-*r/ 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-*r/ 97.8%

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

      5. metadata-eval 97.8%

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

   10. Simplified 97.8%

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

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

   1. Initial program 98.5%

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

      1. div-sub 98.8%

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

      2. clear-num 98.4%

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

      3. sub-neg 98.4%

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

      4. div-inv 98.4%

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

      5. clear-num 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. sub-neg 98.7%

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

   6. Simplified 98.7%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. 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. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. clear-num 0.0%

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

      6. frac-2neg 0.0%

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

      7. metadata-eval 0.0%

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

      8. fma-define 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. distribute-neg-frac2 0.0%

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

      2. fma-define 0.0%

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

      3. +-commutative 0.0%

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

      4. distribute-neg-in 0.0%

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

      5. metadata-eval 0.0%

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

      6. *-commutative 0.0%

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

      7. distribute-lft-neg-in 0.0%

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

      8. metadata-eval 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in i around 0 99.6%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-/l* 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \color{blue}{\left(i \cdot \frac{0.5 - 0.5 \cdot \frac{1}{n}}{n}\right)} - 0.01 \cdot \frac{1}{n}} \]

       2. cancel-sign-sub-inv 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{\color{blue}{0.5 + \left(-0.5\right) \cdot \frac{1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{-0.5} \cdot \frac{1}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{\frac{-0.5 \cdot 1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 99.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{\color{blue}{-0.5}}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 99.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{\color{blue}{0.01}}{n}} \]

   11. Simplified 99.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (/ (expm1 (* i (+ 1.0 (/ (* i -0.5) n)))) i))))
   (if (<= n -6.5e+131)
     t_0
     (if (<= n 2.55e-9)
       (/
        -1.0
        (+
         (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n))
         (* 0.01 (/ -1.0 n))))
       (if (<= n 1.3e+103)
         (*
          n
          (/
           (*
            i
            (+
             100.0
             (*
              i
              (*
               100.0
               (+
                (*
                 i
                 (+
                  0.16666666666666666
                  (- (/ 0.3333333333333333 (pow n 2.0)) (/ 0.5 n))))
                (- 0.5 (/ 0.5 n)))))))
           i))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (expm1((i * (1.0 + ((i * -0.5) / n)))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 1.3e+103) {
		tmp = n * ((i * (100.0 + (i * (100.0 * ((i * (0.16666666666666666 + ((0.3333333333333333 / pow(n, 2.0)) - (0.5 / n)))) + (0.5 - (0.5 / n))))))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) * (Math.expm1((i * (1.0 + ((i * -0.5) / n)))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 1.3e+103) {
		tmp = n * ((i * (100.0 + (i * (100.0 * ((i * (0.16666666666666666 + ((0.3333333333333333 / Math.pow(n, 2.0)) - (0.5 / n)))) + (0.5 - (0.5 / n))))))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (math.expm1((i * (1.0 + ((i * -0.5) / n)))) / i)
	tmp = 0
	if n <= -6.5e+131:
		tmp = t_0
	elif n <= 2.55e-9:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	elif n <= 1.3e+103:
		tmp = n * ((i * (100.0 + (i * (100.0 * ((i * (0.16666666666666666 + ((0.3333333333333333 / math.pow(n, 2.0)) - (0.5 / n)))) + (0.5 - (0.5 / n))))))) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(expm1(Float64(i * Float64(1.0 + Float64(Float64(i * -0.5) / n)))) / i))
	tmp = 0.0
	if (n <= -6.5e+131)
		tmp = t_0;
	elseif (n <= 2.55e-9)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	elseif (n <= 1.3e+103)
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(i * Float64(100.0 * Float64(Float64(i * Float64(0.16666666666666666 + Float64(Float64(0.3333333333333333 / (n ^ 2.0)) - Float64(0.5 / n)))) + Float64(0.5 - Float64(0.5 / n))))))) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[N[(i * N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e+131], t$95$0, If[LessEqual[n, 2.55e-9], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.3e+103], N[(n * N[(N[(i * N[(100.0 + N[(i * N[(100.0 * N[(N[(i * N[(0.16666666666666666 + N[(N[(0.3333333333333333 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.5e131 or 1.3000000000000001e103 < n

   1. Initial program 14.2%

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

      1. associate-*r/ 14.2%

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

      2. sub-neg 14.2%

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

      3. distribute-rgt-in 14.2%

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

      4. metadata-eval 14.2%

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

      5. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

      1. metadata-eval 14.2%

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

      2. metadata-eval 14.2%

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

      3. distribute-rgt-in 14.2%

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

      4. sub-neg 14.2%

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

      5. *-commutative 14.2%

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

      6. associate-*l/ 14.2%

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

      7. associate-/r/ 14.9%

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

      8. associate-*l* 14.9%

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

      9. add-exp-log 14.9%

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

      10. expm1-define 14.9%

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

      11. log-pow 12.1%

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

      12. log1p-define 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in i around 0 88.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + -0.5 \cdot \frac{i}{n}\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \color{blue}{\frac{-0.5 \cdot i}{n}}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{\color{blue}{i \cdot -0.5}}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

   9. Simplified 88.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

   if -6.5e131 < n < 2.55000000000000009e-9

   1. Initial program 35.0%

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

      1. associate-/r/ 35.2%

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

      2. associate-*r* 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-*r/ 35.2%

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

      5. sub-neg 35.2%

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

      6. distribute-lft-in 35.2%

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

      7. metadata-eval 35.2%

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

      8. metadata-eval 35.2%

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

      9. metadata-eval 35.2%

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

      10. fma-define 35.2%

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

      11. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. *-commutative 35.2%

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

      2. fma-undefine 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-/r/ 35.0%

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

      5. clear-num 35.0%

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

      6. frac-2neg 35.0%

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

      7. metadata-eval 35.0%

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

      8. fma-define 35.0%

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

   6. Applied egg-rr 35.0%

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

      1. distribute-neg-frac2 35.0%

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

      2. fma-define 35.0%

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

      3. +-commutative 35.0%

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

      4. distribute-neg-in 35.0%

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

      5. metadata-eval 35.0%

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

      6. *-commutative 35.0%

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

      7. distribute-lft-neg-in 35.0%

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

      8. metadata-eval 35.0%

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

   8. Simplified 35.0%

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

   9. Taylor expanded in i around 0 80.2%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

   if 2.55000000000000009e-9 < n < 1.3000000000000001e103

   1. Initial program 42.3%

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

      1. associate-/r/ 42.5%

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

      2. associate-*r* 42.5%

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

      3. *-commutative 42.5%

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

      4. associate-*r/ 42.5%

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

      5. sub-neg 42.5%

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

      6. distribute-lft-in 42.5%

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

      7. metadata-eval 42.5%

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

      8. metadata-eval 42.5%

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

      9. metadata-eval 42.5%

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

      10. fma-define 42.5%

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

      11. metadata-eval 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in i around 0 91.1%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}}{i} \]
   6. Step-by-step derivation

      1. distribute-lft-out 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \color{blue}{\left(100 \cdot \left(i \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right)}{i} \]

      2. associate--l+ 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \color{blue}{\left(0.16666666666666666 + \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      3. associate-*r/ 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      4. metadata-eval 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      5. associate-*r/ 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      6. metadata-eval 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      7. associate-*r/ 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right)}{i} \]

      8. metadata-eval 91.1%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right)}{i} \]

   7. Simplified 91.1%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)}}{i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+101}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (/ (expm1 (* i (+ 1.0 (/ (* i -0.5) n)))) i))))
   (if (<= n -6.5e+131)
     t_0
     (if (<= n 2.55e-9)
       (/
        -1.0
        (+
         (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n))
         (* 0.01 (/ -1.0 n))))
       (if (<= n 2.7e+101)
         (*
          n
          (+
           100.0
           (*
            i
            (*
             100.0
             (+
              (*
               i
               (+
                0.16666666666666666
                (- (/ 0.3333333333333333 (pow n 2.0)) (/ 0.5 n))))
              (- 0.5 (/ 0.5 n)))))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (expm1((i * (1.0 + ((i * -0.5) / n)))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 2.7e+101) {
		tmp = n * (100.0 + (i * (100.0 * ((i * (0.16666666666666666 + ((0.3333333333333333 / pow(n, 2.0)) - (0.5 / n)))) + (0.5 - (0.5 / n))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) * (Math.expm1((i * (1.0 + ((i * -0.5) / n)))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 2.7e+101) {
		tmp = n * (100.0 + (i * (100.0 * ((i * (0.16666666666666666 + ((0.3333333333333333 / Math.pow(n, 2.0)) - (0.5 / n)))) + (0.5 - (0.5 / n))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (math.expm1((i * (1.0 + ((i * -0.5) / n)))) / i)
	tmp = 0
	if n <= -6.5e+131:
		tmp = t_0
	elif n <= 2.55e-9:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	elif n <= 2.7e+101:
		tmp = n * (100.0 + (i * (100.0 * ((i * (0.16666666666666666 + ((0.3333333333333333 / math.pow(n, 2.0)) - (0.5 / n)))) + (0.5 - (0.5 / n))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(expm1(Float64(i * Float64(1.0 + Float64(Float64(i * -0.5) / n)))) / i))
	tmp = 0.0
	if (n <= -6.5e+131)
		tmp = t_0;
	elseif (n <= 2.55e-9)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	elseif (n <= 2.7e+101)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(100.0 * Float64(Float64(i * Float64(0.16666666666666666 + Float64(Float64(0.3333333333333333 / (n ^ 2.0)) - Float64(0.5 / n)))) + Float64(0.5 - Float64(0.5 / n)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[N[(i * N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e+131], t$95$0, If[LessEqual[n, 2.55e-9], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.7e+101], N[(n * N[(100.0 + N[(i * N[(100.0 * N[(N[(i * N[(0.16666666666666666 + N[(N[(0.3333333333333333 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.5e131 or 2.70000000000000006e101 < n

   1. Initial program 14.2%

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

      1. associate-*r/ 14.2%

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

      2. sub-neg 14.2%

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

      3. distribute-rgt-in 14.2%

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

      4. metadata-eval 14.2%

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

      5. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

      1. metadata-eval 14.2%

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

      2. metadata-eval 14.2%

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

      3. distribute-rgt-in 14.2%

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

      4. sub-neg 14.2%

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

      5. *-commutative 14.2%

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

      6. associate-*l/ 14.2%

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

      7. associate-/r/ 14.9%

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

      8. associate-*l* 14.9%

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

      9. add-exp-log 14.9%

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

      10. expm1-define 14.9%

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

      11. log-pow 12.1%

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

      12. log1p-define 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in i around 0 88.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + -0.5 \cdot \frac{i}{n}\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \color{blue}{\frac{-0.5 \cdot i}{n}}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{\color{blue}{i \cdot -0.5}}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

   9. Simplified 88.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

   if -6.5e131 < n < 2.55000000000000009e-9

   1. Initial program 35.0%

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

      1. associate-/r/ 35.2%

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

      2. associate-*r* 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-*r/ 35.2%

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

      5. sub-neg 35.2%

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

      6. distribute-lft-in 35.2%

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

      7. metadata-eval 35.2%

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

      8. metadata-eval 35.2%

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

      9. metadata-eval 35.2%

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

      10. fma-define 35.2%

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

      11. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. *-commutative 35.2%

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

      2. fma-undefine 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-/r/ 35.0%

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

      5. clear-num 35.0%

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

      6. frac-2neg 35.0%

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

      7. metadata-eval 35.0%

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

      8. fma-define 35.0%

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

   6. Applied egg-rr 35.0%

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

      1. distribute-neg-frac2 35.0%

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

      2. fma-define 35.0%

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

      3. +-commutative 35.0%

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

      4. distribute-neg-in 35.0%

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

      5. metadata-eval 35.0%

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

      6. *-commutative 35.0%

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

      7. distribute-lft-neg-in 35.0%

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

      8. metadata-eval 35.0%

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

   8. Simplified 35.0%

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

   9. Taylor expanded in i around 0 80.2%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

   if 2.55000000000000009e-9 < n < 2.70000000000000006e101

   1. Initial program 42.3%

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

      1. associate-/r/ 42.5%

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

      2. associate-*r* 42.5%

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

      3. *-commutative 42.5%

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

      4. associate-*r/ 42.5%

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

      5. sub-neg 42.5%

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

      6. distribute-lft-in 42.5%

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

      7. metadata-eval 42.5%

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

      8. metadata-eval 42.5%

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

      9. metadata-eval 42.5%

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

      10. fma-define 42.5%

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

      11. metadata-eval 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in i around 0 87.2%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \color{blue}{\left(100 \cdot \left(i \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      2. associate--l+ 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \color{blue}{\left(0.16666666666666666 + \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      3. associate-*r/ 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      4. metadata-eval 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      5. associate-*r/ 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      6. metadata-eval 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. associate-*r/ 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right) \]

      8. metadata-eval 87.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right) \]

   7. Simplified 87.2%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+101}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{+103}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (/ (expm1 (* i (+ 1.0 (/ (* i -0.5) n)))) i))))
   (if (<= n -6.5e+131)
     t_0
     (if (<= n 2.55e-9)
       (/
        -1.0
        (+
         (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n))
         (* 0.01 (/ -1.0 n))))
       (if (<= n 5.6e+103)
         (* n (/ (* i (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n)))))) i))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (expm1((i * (1.0 + ((i * -0.5) / n)))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 5.6e+103) {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) * (Math.expm1((i * (1.0 + ((i * -0.5) / n)))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 5.6e+103) {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (math.expm1((i * (1.0 + ((i * -0.5) / n)))) / i)
	tmp = 0
	if n <= -6.5e+131:
		tmp = t_0
	elif n <= 2.55e-9:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	elif n <= 5.6e+103:
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(expm1(Float64(i * Float64(1.0 + Float64(Float64(i * -0.5) / n)))) / i))
	tmp = 0.0
	if (n <= -6.5e+131)
		tmp = t_0;
	elseif (n <= 2.55e-9)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	elseif (n <= 5.6e+103)
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n)))))) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[N[(i * N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e+131], t$95$0, If[LessEqual[n, 2.55e-9], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.6e+103], N[(n * N[(N[(i * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.5e131 or 5.60000000000000017e103 < n

   1. Initial program 14.2%

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

      1. associate-*r/ 14.2%

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

      2. sub-neg 14.2%

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

      3. distribute-rgt-in 14.2%

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

      4. metadata-eval 14.2%

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

      5. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

      1. metadata-eval 14.2%

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

      2. metadata-eval 14.2%

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

      3. distribute-rgt-in 14.2%

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

      4. sub-neg 14.2%

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

      5. *-commutative 14.2%

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

      6. associate-*l/ 14.2%

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

      7. associate-/r/ 14.9%

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

      8. associate-*l* 14.9%

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

      9. add-exp-log 14.9%

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

      10. expm1-define 14.9%

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

      11. log-pow 12.1%

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

      12. log1p-define 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in i around 0 88.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + -0.5 \cdot \frac{i}{n}\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \color{blue}{\frac{-0.5 \cdot i}{n}}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{\color{blue}{i \cdot -0.5}}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

   9. Simplified 88.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

   if -6.5e131 < n < 2.55000000000000009e-9

   1. Initial program 35.0%

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

      1. associate-/r/ 35.2%

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

      2. associate-*r* 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-*r/ 35.2%

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

      5. sub-neg 35.2%

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

      6. distribute-lft-in 35.2%

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

      7. metadata-eval 35.2%

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

      8. metadata-eval 35.2%

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

      9. metadata-eval 35.2%

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

      10. fma-define 35.2%

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

      11. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. *-commutative 35.2%

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

      2. fma-undefine 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-/r/ 35.0%

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

      5. clear-num 35.0%

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

      6. frac-2neg 35.0%

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

      7. metadata-eval 35.0%

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

      8. fma-define 35.0%

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

   6. Applied egg-rr 35.0%

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

      1. distribute-neg-frac2 35.0%

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

      2. fma-define 35.0%

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

      3. +-commutative 35.0%

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

      4. distribute-neg-in 35.0%

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

      5. metadata-eval 35.0%

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

      6. *-commutative 35.0%

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

      7. distribute-lft-neg-in 35.0%

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

      8. metadata-eval 35.0%

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

   8. Simplified 35.0%

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

   9. Taylor expanded in i around 0 80.2%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

   if 2.55000000000000009e-9 < n < 5.60000000000000017e103

   1. Initial program 42.3%

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

      1. associate-/r/ 42.5%

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

      2. associate-*r* 42.5%

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

      3. *-commutative 42.5%

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

      4. associate-*r/ 42.5%

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

      5. sub-neg 42.5%

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

      6. distribute-lft-in 42.5%

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

      7. metadata-eval 42.5%

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

      8. metadata-eval 42.5%

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

      9. metadata-eval 42.5%

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

      10. fma-define 42.5%

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

      11. metadata-eval 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in i around 0 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-*r/ 86.8%

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

      3. metadata-eval 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{+103}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 (* i (+ 1.0 (/ (* i -0.5) n))))) i))))
   (if (<= n -6.5e+131)
     t_0
     (if (<= n 2.55e-9)
       (/
        -1.0
        (+
         (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n))
         (* 0.01 (/ -1.0 n))))
       (if (<= n 1.15e+99)
         (* n (/ (* i (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n)))))) i))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1((i * (1.0 + ((i * -0.5) / n))))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 1.15e+99) {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 * Math.expm1((i * (1.0 + ((i * -0.5) / n))))) / i);
	double tmp;
	if (n <= -6.5e+131) {
		tmp = t_0;
	} else if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else if (n <= 1.15e+99) {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 * math.expm1((i * (1.0 + ((i * -0.5) / n))))) / i)
	tmp = 0
	if n <= -6.5e+131:
		tmp = t_0
	elif n <= 2.55e-9:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	elif n <= 1.15e+99:
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(Float64(i * Float64(1.0 + Float64(Float64(i * -0.5) / n))))) / i))
	tmp = 0.0
	if (n <= -6.5e+131)
		tmp = t_0;
	elseif (n <= 2.55e-9)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	elseif (n <= 1.15e+99)
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n)))))) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[N[(i * N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e+131], t$95$0, If[LessEqual[n, 2.55e-9], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e+99], N[(n * N[(N[(i * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.5e131 or 1.1500000000000001e99 < n

   1. Initial program 14.2%

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

      1. associate-/r/ 14.9%

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

      2. associate-*r* 14.9%

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

      3. *-commutative 14.9%

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

      4. associate-*r/ 14.9%

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

      5. sub-neg 14.9%

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

      6. distribute-lft-in 14.9%

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

      7. metadata-eval 14.9%

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

      8. metadata-eval 14.9%

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

      9. metadata-eval 14.9%

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

      10. fma-define 14.9%

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

      11. metadata-eval 14.9%

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

   3. Simplified 14.9%

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

      1. fma-undefine 14.9%

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

      2. metadata-eval 14.9%

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

      3. metadata-eval 14.9%

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

      4. distribute-lft-in 14.9%

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

      5. sub-neg 14.9%

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

      6. *-commutative 14.9%

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

      7. add-exp-log 14.9%

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

      8. expm1-define 14.9%

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

      9. log-pow 12.1%

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

      10. log1p-define 62.1%

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

   6. Applied egg-rr 62.1%

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

   7. Taylor expanded in i around 0 88.5%

      \[\leadsto n \cdot \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + -0.5 \cdot \frac{i}{n}\right)}\right) \cdot 100}{i} \]
   8. Step-by-step derivation

      1. associate-*r/ 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \color{blue}{\frac{-0.5 \cdot i}{n}}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative 88.7%

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \left(1 + \frac{\color{blue}{i \cdot -0.5}}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

   9. Simplified 88.5%

      \[\leadsto n \cdot \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)}\right) \cdot 100}{i} \]

   if -6.5e131 < n < 2.55000000000000009e-9

   1. Initial program 35.0%

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

      1. associate-/r/ 35.2%

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

      2. associate-*r* 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-*r/ 35.2%

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

      5. sub-neg 35.2%

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

      6. distribute-lft-in 35.2%

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

      7. metadata-eval 35.2%

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

      8. metadata-eval 35.2%

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

      9. metadata-eval 35.2%

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

      10. fma-define 35.2%

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

      11. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. *-commutative 35.2%

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

      2. fma-undefine 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-/r/ 35.0%

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

      5. clear-num 35.0%

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

      6. frac-2neg 35.0%

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

      7. metadata-eval 35.0%

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

      8. fma-define 35.0%

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

   6. Applied egg-rr 35.0%

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

      1. distribute-neg-frac2 35.0%

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

      2. fma-define 35.0%

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

      3. +-commutative 35.0%

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

      4. distribute-neg-in 35.0%

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

      5. metadata-eval 35.0%

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

      6. *-commutative 35.0%

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

      7. distribute-lft-neg-in 35.0%

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

      8. metadata-eval 35.0%

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

   8. Simplified 35.0%

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

   9. Taylor expanded in i around 0 80.2%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

   if 2.55000000000000009e-9 < n < 1.1500000000000001e99

   1. Initial program 42.3%

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

      1. associate-/r/ 42.5%

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

      2. associate-*r* 42.5%

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

      3. *-commutative 42.5%

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

      4. associate-*r/ 42.5%

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

      5. sub-neg 42.5%

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

      6. distribute-lft-in 42.5%

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

      7. metadata-eval 42.5%

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

      8. metadata-eval 42.5%

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

      9. metadata-eval 42.5%

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

      10. fma-define 42.5%

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

      11. metadata-eval 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in i around 0 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-*r/ 86.8%

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

      3. metadata-eval 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{+131}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i \cdot \left(1 + \frac{i \cdot -0.5}{n}\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{0.5}{n}\\ \mathbf{if}\;n \leq -7.8 \cdot 10^{-66}:\\ \;\;\;\;100 \cdot \left(n + t\_0 \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot t\_0\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ 0.5 n))))
   (if (<= n -7.8e-66)
     (* 100.0 (+ n (* t_0 (* i n))))
     (if (<= n 2.5e-9)
       (* 100.0 (/ i (/ i n)))
       (* n (/ (* i (+ 100.0 (* 100.0 (* i t_0)))) i))))))

C

double code(double i, double n) {
	double t_0 = 0.5 - (0.5 / n);
	double tmp;
	if (n <= -7.8e-66) {
		tmp = 100.0 * (n + (t_0 * (i * n)));
	} else if (n <= 2.5e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * (100.0 + (100.0 * (i * t_0)))) / i);
	}
	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 = 0.5d0 - (0.5d0 / n)
    if (n <= (-7.8d-66)) then
        tmp = 100.0d0 * (n + (t_0 * (i * n)))
    else if (n <= 2.5d-9) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * ((i * (100.0d0 + (100.0d0 * (i * t_0)))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 0.5 - (0.5 / n);
	double tmp;
	if (n <= -7.8e-66) {
		tmp = 100.0 * (n + (t_0 * (i * n)));
	} else if (n <= 2.5e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * (100.0 + (100.0 * (i * t_0)))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 0.5 - (0.5 / n)
	tmp = 0
	if n <= -7.8e-66:
		tmp = 100.0 * (n + (t_0 * (i * n)))
	elif n <= 2.5e-9:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * ((i * (100.0 + (100.0 * (i * t_0)))) / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(0.5 - Float64(0.5 / n))
	tmp = 0.0
	if (n <= -7.8e-66)
		tmp = Float64(100.0 * Float64(n + Float64(t_0 * Float64(i * n))));
	elseif (n <= 2.5e-9)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(100.0 * Float64(i * t_0)))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 0.5 - (0.5 / n);
	tmp = 0.0;
	if (n <= -7.8e-66)
		tmp = 100.0 * (n + (t_0 * (i * n)));
	elseif (n <= 2.5e-9)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * ((i * (100.0 + (100.0 * (i * t_0)))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7.8e-66], N[(100.0 * N[(n + N[(t$95$0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-9], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * N[(100.0 + N[(100.0 * N[(i * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.79999999999999965e-66

   1. Initial program 22.3%

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

   3. Taylor expanded in i around 0 64.7%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

      3. associate-*r/ 64.7%

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

      4. metadata-eval 64.7%

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

   5. Simplified 64.7%

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

   if -7.79999999999999965e-66 < n < 2.5000000000000001e-9

   1. Initial program 39.2%

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

   3. Taylor expanded in i around 0 40.2%

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

      1. +-commutative 40.2%

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

   5. Simplified 40.2%

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

   6. Taylor expanded in i around 0 66.7%

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

   if 2.5000000000000001e-9 < n

   1. Initial program 21.4%

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

      1. associate-/r/ 22.0%

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

      2. associate-*r* 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-*r/ 22.0%

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

      5. sub-neg 22.0%

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

      6. distribute-lft-in 22.0%

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

      7. metadata-eval 22.0%

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

      8. metadata-eval 22.0%

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

      9. metadata-eval 22.0%

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

      10. fma-define 22.0%

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

      11. metadata-eval 22.0%

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

   3. Simplified 22.0%

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

   5. Taylor expanded in i around 0 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-*r/ 71.5%

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

      3. metadata-eval 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{-66}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.55e-9)
   (/
    -1.0
    (+ (* 0.01 (/ (* i (+ 0.5 (* 0.5 (/ -1.0 n)))) n)) (* 0.01 (/ -1.0 n))))
   (* n (/ (* i (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n)))))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 2.55d-9) then
        tmp = (-1.0d0) / ((0.01d0 * ((i * (0.5d0 + (0.5d0 * ((-1.0d0) / n)))) / n)) + (0.01d0 * ((-1.0d0) / n)))
    else
        tmp = n * ((i * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 2.55e-9) {
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	} else {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 2.55e-9:
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)))
	else:
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.55e-9)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))) / n)) + Float64(0.01 * Float64(-1.0 / n))));
	else
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n)))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 2.55e-9)
		tmp = -1.0 / ((0.01 * ((i * (0.5 + (0.5 * (-1.0 / n)))) / n)) + (0.01 * (-1.0 / n)));
	else
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.55e-9], N[(-1.0 / N[(N[(0.01 * N[(N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.55000000000000009e-9

   1. Initial program 29.5%

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

      1. associate-/r/ 29.9%

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

      2. associate-*r* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r/ 29.9%

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

      5. sub-neg 29.9%

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

      6. distribute-lft-in 29.9%

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

      7. metadata-eval 29.9%

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

      8. metadata-eval 29.9%

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

      9. metadata-eval 29.9%

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

      10. fma-define 29.9%

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

      11. metadata-eval 29.9%

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

   3. Simplified 29.9%

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

      1. *-commutative 29.9%

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

      2. fma-undefine 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-/r/ 29.5%

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

      5. clear-num 29.5%

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

      6. frac-2neg 29.5%

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

      7. metadata-eval 29.5%

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

      8. fma-define 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. distribute-neg-frac2 29.5%

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

      2. fma-define 29.5%

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

      3. +-commutative 29.5%

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

      4. distribute-neg-in 29.5%

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

      5. metadata-eval 29.5%

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

      6. *-commutative 29.5%

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

      7. distribute-lft-neg-in 29.5%

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

      8. metadata-eval 29.5%

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

   8. Simplified 29.5%

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

   9. Taylor expanded in i around 0 75.5%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]

   if 2.55000000000000009e-9 < n

   1. Initial program 21.4%

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

      1. associate-/r/ 22.0%

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

      2. associate-*r* 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-*r/ 22.0%

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

      5. sub-neg 22.0%

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

      6. distribute-lft-in 22.0%

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

      7. metadata-eval 22.0%

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

      8. metadata-eval 22.0%

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

      9. metadata-eval 22.0%

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

      10. fma-define 22.0%

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

      11. metadata-eval 22.0%

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

   3. Simplified 22.0%

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

   5. Taylor expanded in i around 0 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-*r/ 71.5%

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

      3. metadata-eval 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \frac{i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)}{n} + 0.01 \cdot \frac{-1}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.3e-9)
   (/ -1.0 (- (* 0.01 (* i (/ (+ 0.5 (/ -0.5 n)) n))) (/ 0.01 n)))
   (* n (/ (* i (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n)))))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.3e-9) {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	} else {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 2.3d-9) then
        tmp = (-1.0d0) / ((0.01d0 * (i * ((0.5d0 + ((-0.5d0) / n)) / n))) - (0.01d0 / n))
    else
        tmp = n * ((i * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 2.3e-9) {
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	} else {
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 2.3e-9:
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n))
	else:
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.3e-9)
		tmp = Float64(-1.0 / Float64(Float64(0.01 * Float64(i * Float64(Float64(0.5 + Float64(-0.5 / n)) / n))) - Float64(0.01 / n)));
	else
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n)))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 2.3e-9)
		tmp = -1.0 / ((0.01 * (i * ((0.5 + (-0.5 / n)) / n))) - (0.01 / n));
	else
		tmp = n * ((i * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.3e-9], N[(-1.0 / N[(N[(0.01 * N[(i * N[(N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.2999999999999999e-9

   1. Initial program 29.5%

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

      1. associate-/r/ 29.9%

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

      2. associate-*r* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r/ 29.9%

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

      5. sub-neg 29.9%

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

      6. distribute-lft-in 29.9%

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

      7. metadata-eval 29.9%

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

      8. metadata-eval 29.9%

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

      9. metadata-eval 29.9%

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

      10. fma-define 29.9%

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

      11. metadata-eval 29.9%

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

   3. Simplified 29.9%

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

      1. *-commutative 29.9%

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

      2. fma-undefine 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-/r/ 29.5%

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

      5. clear-num 29.5%

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

      6. frac-2neg 29.5%

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

      7. metadata-eval 29.5%

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

      8. fma-define 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. distribute-neg-frac2 29.5%

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

      2. fma-define 29.5%

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

      3. +-commutative 29.5%

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

      4. distribute-neg-in 29.5%

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

      5. metadata-eval 29.5%

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

      6. *-commutative 29.5%

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

      7. distribute-lft-neg-in 29.5%

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

      8. metadata-eval 29.5%

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

   8. Simplified 29.5%

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

   9. Taylor expanded in i around 0 75.5%

      \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \frac{i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{n} - 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-/l* 71.6%

          \[\leadsto \frac{-1}{0.01 \cdot \color{blue}{\left(i \cdot \frac{0.5 - 0.5 \cdot \frac{1}{n}}{n}\right)} - 0.01 \cdot \frac{1}{n}} \]

       2. cancel-sign-sub-inv 71.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{\color{blue}{0.5 + \left(-0.5\right) \cdot \frac{1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 71.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{-0.5} \cdot \frac{1}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 71.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \color{blue}{\frac{-0.5 \cdot 1}{n}}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 71.6%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{\color{blue}{-0.5}}{n}}{n}\right) - 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 71.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 71.7%

          \[\leadsto \frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{\color{blue}{0.01}}{n}} \]

   11. Simplified 71.7%

       \[\leadsto \frac{-1}{\color{blue}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}} \]

   if 2.2999999999999999e-9 < n

   1. Initial program 21.4%

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

      1. associate-/r/ 22.0%

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

      2. associate-*r* 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-*r/ 22.0%

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

      5. sub-neg 22.0%

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

      6. distribute-lft-in 22.0%

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

      7. metadata-eval 22.0%

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

      8. metadata-eval 22.0%

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

      9. metadata-eval 22.0%

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

      10. fma-define 22.0%

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

      11. metadata-eval 22.0%

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

   3. Simplified 22.0%

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

   5. Taylor expanded in i around 0 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-*r/ 71.5%

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

      3. metadata-eval 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{0.01 \cdot \left(i \cdot \frac{0.5 + \frac{-0.5}{n}}{n}\right) - \frac{0.01}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{-76}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.6e-76)
   (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n))))
   (if (<= n 8.2e-38) (* 100.0 (/ i (/ i n))) (* 100.0 (/ (* i n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.6e-76) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (n <= 8.2e-38) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-6.6d-76)) then
        tmp = 100.0d0 * (n + ((0.5d0 - (0.5d0 / n)) * (i * n)))
    else if (n <= 8.2d-38) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((i * n) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6.6e-76) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (n <= 8.2e-38) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6.6e-76:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	elif n <= 8.2e-38:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((i * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.6e-76)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * n))));
	elseif (n <= 8.2e-38)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -6.6e-76)
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	elseif (n <= 8.2e-38)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((i * n) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.6e-76], N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.2e-38], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.59999999999999967e-76

   1. Initial program 22.3%

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

   3. Taylor expanded in i around 0 64.7%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

      3. associate-*r/ 64.7%

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

      4. metadata-eval 64.7%

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

   5. Simplified 64.7%

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

   if -6.59999999999999967e-76 < n < 8.1999999999999996e-38

   1. Initial program 41.9%

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

   3. Taylor expanded in i around 0 43.9%

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

      1. +-commutative 43.9%

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

   5. Simplified 43.9%

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

   6. Taylor expanded in i around 0 67.4%

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

   if 8.1999999999999996e-38 < n

   1. Initial program 20.7%

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

   3. Taylor expanded in i around 0 3.6%

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

      1. +-commutative 3.6%

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

   5. Simplified 3.6%

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

      1. clear-num 3.6%

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

      2. inv-pow 3.6%

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

   7. Applied egg-rr 3.6%

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

      1. unpow-1 3.6%

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

   9. Simplified 3.6%

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

       1. clear-num 3.6%

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

       2. div-inv 3.6%

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

       3. add-exp-log 2.3%

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

       4. expm1-define 2.3%

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

       5. +-commutative 2.3%

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

       6. log1p-define 27.4%

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

       7. expm1-log1p-u 28.6%

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

       8. clear-num 28.7%

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

   11. Applied egg-rr 28.7%

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

       1. associate-*r/ 67.0%

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

   13. Applied egg-rr 67.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{-76}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+21} \lor \neg \left(n \leq 8.5 \cdot 10^{-41}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1e+21) (not (<= n 8.5e-41)))
   (* 100.0 (/ (* i n) i))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1e+21) || !(n <= 8.5e-41)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1d+21)) .or. (.not. (n <= 8.5d-41))) then
        tmp = 100.0d0 * ((i * n) / i)
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1e+21) || !(n <= 8.5e-41)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1e+21) or not (n <= 8.5e-41):
		tmp = 100.0 * ((i * n) / i)
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1e+21) || !(n <= 8.5e-41))
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1e+21) || ~((n <= 8.5e-41)))
		tmp = 100.0 * ((i * n) / i);
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1e+21], N[Not[LessEqual[n, 8.5e-41]], $MachinePrecision]], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1e21 or 8.4999999999999996e-41 < n

   1. Initial program 22.1%

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

   3. Taylor expanded in i around 0 3.7%

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

      1. +-commutative 3.7%

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

   5. Simplified 3.7%

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

      1. clear-num 3.6%

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

      2. inv-pow 3.6%

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

   7. Applied egg-rr 3.6%

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

      1. unpow-1 3.6%

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

   9. Simplified 3.6%

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

       1. clear-num 3.7%

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

       2. div-inv 3.6%

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

       3. add-exp-log 2.4%

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

       4. expm1-define 2.4%

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

       5. +-commutative 2.4%

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

       6. log1p-define 28.1%

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

       7. expm1-log1p-u 29.3%

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

       8. clear-num 29.4%

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

   11. Applied egg-rr 29.4%

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

       1. associate-*r/ 66.0%

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

   13. Applied egg-rr 66.0%

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

   if -1e21 < n < 8.4999999999999996e-41

   1. Initial program 37.0%

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

   3. Taylor expanded in i around 0 36.3%

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

      1. +-commutative 36.3%

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

   5. Simplified 36.3%

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

   6. Taylor expanded in i around 0 66.6%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+21} \lor \neg \left(n \leq 8.5 \cdot 10^{-41}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -9 \cdot 10^{+151} \lor \neg \left(i \leq 5 \cdot 10^{-50}\right):\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -9e+151) (not (<= i 5e-50)))
   (* 100.0 (* i (/ n i)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -9e+151) || !(i <= 5e-50)) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-9d+151)) .or. (.not. (i <= 5d-50))) then
        tmp = 100.0d0 * (i * (n / i))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -9e+151) || !(i <= 5e-50)) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -9e+151) or not (i <= 5e-50):
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -9e+151) || !(i <= 5e-50))
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -9e+151) || ~((i <= 5e-50)))
		tmp = 100.0 * (i * (n / i));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -9e+151], N[Not[LessEqual[i, 5e-50]], $MachinePrecision]], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -8.9999999999999997e151 or 4.99999999999999968e-50 < i

   1. Initial program 56.5%

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

   3. Taylor expanded in i around 0 26.5%

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

      1. +-commutative 26.5%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 26.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. clear-num 26.5%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

      2. inv-pow 26.5%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]

   7. Applied egg-rr 26.5%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]
   8. Step-by-step derivation

      1. unpow-1 26.5%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

   9. Simplified 26.5%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   10. Step-by-step derivation

       1. clear-num 26.5%

          \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{i}{n}}} \]

       2. div-inv 26.5%

          \[\leadsto 100 \cdot \color{blue}{\left(\left(\left(i + 1\right) - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \]

       3. add-exp-log 13.6%

          \[\leadsto 100 \cdot \left(\left(\color{blue}{e^{\log \left(i + 1\right)}} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       4. expm1-define 13.6%

          \[\leadsto 100 \cdot \left(\color{blue}{\mathsf{expm1}\left(\log \left(i + 1\right)\right)} \cdot \frac{1}{\frac{i}{n}}\right) \]

       5. +-commutative 13.6%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       6. log1p-define 20.8%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       7. expm1-log1p-u 33.7%

          \[\leadsto 100 \cdot \left(\color{blue}{i} \cdot \frac{1}{\frac{i}{n}}\right) \]

       8. clear-num 32.7%

          \[\leadsto 100 \cdot \left(i \cdot \color{blue}{\frac{n}{i}}\right) \]

   11. Applied egg-rr 32.7%

       \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \frac{n}{i}\right)} \]

   if -8.9999999999999997e151 < i < 4.99999999999999968e-50

   1. Initial program 8.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 7.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 7.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 75.9%

      \[\leadsto \color{blue}{100 \cdot n} \]
   7. Step-by-step derivation

      1. *-commutative 75.9%

         \[\leadsto \color{blue}{n \cdot 100} \]

   8. Simplified 75.9%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -9 \cdot 10^{+151} \lor \neg \left(i \leq 5 \cdot 10^{-50}\right):\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{+21}:\\ \;\;\;\;100 \cdot \frac{1}{i \cdot \frac{\frac{1}{n}}{i}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-47}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.7e+21)
   (* 100.0 (/ 1.0 (* i (/ (/ 1.0 n) i))))
   (if (<= n 5e-47) (* 100.0 (/ i (/ i n))) (* 100.0 (/ (* i n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.7e+21) {
		tmp = 100.0 * (1.0 / (i * ((1.0 / n) / i)));
	} else if (n <= 5e-47) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.7d+21)) then
        tmp = 100.0d0 * (1.0d0 / (i * ((1.0d0 / n) / i)))
    else if (n <= 5d-47) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((i * n) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.7e+21) {
		tmp = 100.0 * (1.0 / (i * ((1.0 / n) / i)));
	} else if (n <= 5e-47) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.7e+21:
		tmp = 100.0 * (1.0 / (i * ((1.0 / n) / i)))
	elif n <= 5e-47:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((i * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.7e+21)
		tmp = Float64(100.0 * Float64(1.0 / Float64(i * Float64(Float64(1.0 / n) / i))));
	elseif (n <= 5e-47)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.7e+21)
		tmp = 100.0 * (1.0 / (i * ((1.0 / n) / i)));
	elseif (n <= 5e-47)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((i * n) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.7e+21], N[(100.0 * N[(1.0 / N[(i * N[(N[(1.0 / n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e-47], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.7e21

   1. Initial program 23.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 3.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative 3.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 3.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. clear-num 3.6%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

      2. inv-pow 3.6%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]

   7. Applied egg-rr 3.6%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]
   8. Step-by-step derivation

      1. unpow-1 3.6%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

   9. Simplified 3.6%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   10. Step-by-step derivation

       1. clear-num 3.8%

          \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{i}{n}}} \]

       2. div-inv 3.6%

          \[\leadsto 100 \cdot \color{blue}{\left(\left(\left(i + 1\right) - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \]

       3. add-exp-log 2.6%

          \[\leadsto 100 \cdot \left(\left(\color{blue}{e^{\log \left(i + 1\right)}} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       4. expm1-define 2.6%

          \[\leadsto 100 \cdot \left(\color{blue}{\mathsf{expm1}\left(\log \left(i + 1\right)\right)} \cdot \frac{1}{\frac{i}{n}}\right) \]

       5. +-commutative 2.6%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       6. log1p-define 28.8%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       7. expm1-log1p-u 29.9%

          \[\leadsto 100 \cdot \left(\color{blue}{i} \cdot \frac{1}{\frac{i}{n}}\right) \]

       8. clear-num 30.0%

          \[\leadsto 100 \cdot \left(i \cdot \color{blue}{\frac{n}{i}}\right) \]

   11. Applied egg-rr 30.0%

       \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \frac{n}{i}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 64.9%

          \[\leadsto 100 \cdot \color{blue}{\frac{i \cdot n}{i}} \]

       2. pow1 64.9%

          \[\leadsto 100 \cdot \frac{i \cdot \color{blue}{{n}^{1}}}{i} \]

       3. metadata-eval 64.9%

          \[\leadsto 100 \cdot \frac{i \cdot {n}^{\color{blue}{\left(--1\right)}}}{i} \]

       4. pow-flip 64.9%

          \[\leadsto 100 \cdot \frac{i \cdot \color{blue}{\frac{1}{{n}^{-1}}}}{i} \]

       5. inv-pow 64.9%

          \[\leadsto 100 \cdot \frac{i \cdot \frac{1}{\color{blue}{\frac{1}{n}}}}{i} \]

       6. div-inv 64.7%

          \[\leadsto 100 \cdot \frac{\color{blue}{\frac{i}{\frac{1}{n}}}}{i} \]

       7. *-un-lft-identity 64.7%

          \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \frac{i}{\frac{1}{n}}}}{i} \]

       8. associate-*l/ 64.7%

          \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{i}{\frac{1}{n}}\right)} \]

       9. *-commutative 64.7%

          \[\leadsto 100 \cdot \color{blue}{\left(\frac{i}{\frac{1}{n}} \cdot \frac{1}{i}\right)} \]

       10. clear-num 64.6%

           \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\frac{\frac{1}{n}}{i}}} \cdot \frac{1}{i}\right) \]

       11. frac-times 65.0%

           \[\leadsto 100 \cdot \color{blue}{\frac{1 \cdot 1}{\frac{\frac{1}{n}}{i} \cdot i}} \]

       12. metadata-eval 65.0%

           \[\leadsto 100 \cdot \frac{\color{blue}{1}}{\frac{\frac{1}{n}}{i} \cdot i} \]

   13. Applied egg-rr 65.0%

       \[\leadsto 100 \cdot \color{blue}{\frac{1}{\frac{\frac{1}{n}}{i} \cdot i}} \]

   if -1.7e21 < n < 5.00000000000000011e-47

   1. Initial program 37.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 36.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative 36.3%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 36.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 66.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 5.00000000000000011e-47 < n

   1. Initial program 20.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 3.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative 3.6%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 3.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. clear-num 3.6%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

      2. inv-pow 3.6%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]

   7. Applied egg-rr 3.6%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]
   8. Step-by-step derivation

      1. unpow-1 3.6%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

   9. Simplified 3.6%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   10. Step-by-step derivation

       1. clear-num 3.6%

          \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{i}{n}}} \]

       2. div-inv 3.6%

          \[\leadsto 100 \cdot \color{blue}{\left(\left(\left(i + 1\right) - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \]

       3. add-exp-log 2.3%

          \[\leadsto 100 \cdot \left(\left(\color{blue}{e^{\log \left(i + 1\right)}} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       4. expm1-define 2.3%

          \[\leadsto 100 \cdot \left(\color{blue}{\mathsf{expm1}\left(\log \left(i + 1\right)\right)} \cdot \frac{1}{\frac{i}{n}}\right) \]

       5. +-commutative 2.3%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       6. log1p-define 27.4%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       7. expm1-log1p-u 28.6%

          \[\leadsto 100 \cdot \left(\color{blue}{i} \cdot \frac{1}{\frac{i}{n}}\right) \]

       8. clear-num 28.7%

          \[\leadsto 100 \cdot \left(i \cdot \color{blue}{\frac{n}{i}}\right) \]

   11. Applied egg-rr 28.7%

       \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \frac{n}{i}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 67.0%

          \[\leadsto 100 \cdot \color{blue}{\frac{i \cdot n}{i}} \]

   13. Applied egg-rr 67.0%

       \[\leadsto 100 \cdot \color{blue}{\frac{i \cdot n}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{+21}:\\ \;\;\;\;100 \cdot \frac{1}{i \cdot \frac{\frac{1}{n}}{i}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-47}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+154}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 10^{-49}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2e+154)
   (* 100.0 (* i (/ n i)))
   (if (<= i 1e-49) (* n 100.0) (* 100.0 (/ i (/ i n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2e+154) {
		tmp = 100.0 * (i * (n / i));
	} else if (i <= 1e-49) {
		tmp = n * 100.0;
	} 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 (i <= (-2d+154)) then
        tmp = 100.0d0 * (i * (n / i))
    else if (i <= 1d-49) then
        tmp = n * 100.0d0
    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 (i <= -2e+154) {
		tmp = 100.0 * (i * (n / i));
	} else if (i <= 1e-49) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2e+154:
		tmp = 100.0 * (i * (n / i))
	elif i <= 1e-49:
		tmp = n * 100.0
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2e+154)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	elseif (i <= 1e-49)
		tmp = Float64(n * 100.0);
	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 (i <= -2e+154)
		tmp = 100.0 * (i * (n / i));
	elseif (i <= 1e-49)
		tmp = n * 100.0;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2e+154], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1e-49], N[(n * 100.0), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.00000000000000007e154

   1. Initial program 79.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 45.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative 45.1%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 45.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. clear-num 45.1%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

      2. inv-pow 45.1%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]

   7. Applied egg-rr 45.1%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{{\left(\frac{n}{i}\right)}^{-1}}} \]
   8. Step-by-step derivation

      1. unpow-1 45.1%

         \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]

   9. Simplified 45.1%

      \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   10. Step-by-step derivation

       1. clear-num 45.1%

          \[\leadsto 100 \cdot \frac{\left(i + 1\right) - 1}{\color{blue}{\frac{i}{n}}} \]

       2. div-inv 45.1%

          \[\leadsto 100 \cdot \color{blue}{\left(\left(\left(i + 1\right) - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \]

       3. add-exp-log 0.0%

          \[\leadsto 100 \cdot \left(\left(\color{blue}{e^{\log \left(i + 1\right)}} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       4. expm1-define 0.0%

          \[\leadsto 100 \cdot \left(\color{blue}{\mathsf{expm1}\left(\log \left(i + 1\right)\right)} \cdot \frac{1}{\frac{i}{n}}\right) \]

       5. +-commutative 0.0%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       6. log1p-define 0.0%

          \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(i\right)}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

       7. expm1-log1p-u 45.1%

          \[\leadsto 100 \cdot \left(\color{blue}{i} \cdot \frac{1}{\frac{i}{n}}\right) \]

       8. clear-num 45.1%

          \[\leadsto 100 \cdot \left(i \cdot \color{blue}{\frac{n}{i}}\right) \]

   11. Applied egg-rr 45.1%

       \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \frac{n}{i}\right)} \]

   if -2.00000000000000007e154 < i < 9.99999999999999936e-50

   1. Initial program 8.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 7.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative 7.0%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 7.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 76.0%

      \[\leadsto \color{blue}{100 \cdot n} \]
   7. Step-by-step derivation

      1. *-commutative 76.0%

         \[\leadsto \color{blue}{n \cdot 100} \]

   8. Simplified 76.0%

      \[\leadsto \color{blue}{n \cdot 100} \]

   if 9.99999999999999936e-50 < i

   1. Initial program 48.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 19.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. +-commutative 19.3%

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   5. Simplified 19.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 28.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 49.2% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * 100.0d0
end function

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 27.0%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0 14.5%

   \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
4. Step-by-step derivation

   1. +-commutative 14.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

5. Simplified 14.5%

   \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + 1\right)} - 1}{\frac{i}{n}} \]

6. Taylor expanded in i around 0 51.7%

   \[\leadsto \color{blue}{100 \cdot n} \]
7. Step-by-step derivation

   1. *-commutative 51.7%

      \[\leadsto \color{blue}{n \cdot 100} \]

8. Simplified 51.7%

   \[\leadsto \color{blue}{n \cdot 100} \]
9. Add Preprocessing

Developer Target 1: 34.1% 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 2024179 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))