Compound Interest

Percentage Accurate: 29.2% → 95.0%

Time: 23.6s

Alternatives: 22

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 29.2% 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: 95.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-46}:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-285}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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-46)
     (* t_1 100.0)
     (if (<= t_1 5e-285)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (/ (+ (* t_0 100.0) -100.0) (/ i n))
         (* n 100.0))))))

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-46) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 5e-285) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-46) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 5e-285) {
		tmp = n * ((100.0 * Math.expm1((n * Math.log1p((i / n))))) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	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-46:
		tmp = t_1 * 100.0
	elif t_1 <= 5e-285:
		tmp = n * ((100.0 * math.expm1((n * math.log1p((i / n))))) / i)
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = n * 100.0
	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-46)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 5e-285)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-46], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 5e-285], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $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)) < -4.99999999999999992e-46

   1. Initial program 100.0%

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

   if -4.99999999999999992e-46 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 5.00000000000000018e-285

   1. Initial program 27.8%

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

      1. associate-/r/ 27.8%

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

      2. associate-*r* 27.8%

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

      3. *-commutative 27.8%

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

      4. associate-*r/ 27.8%

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

      5. sub-neg 27.8%

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

      6. distribute-lft-in 27.9%

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

      7. metadata-eval 27.9%

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

      8. metadata-eval 27.9%

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

      9. metadata-eval 27.9%

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

      10. fma-define 27.8%

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

      11. metadata-eval 27.8%

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

   3. Simplified 27.8%

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

      1. fma-undefine 27.9%

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

      2. metadata-eval 27.9%

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

      3. metadata-eval 27.9%

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

      4. distribute-lft-in 27.8%

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

      5. sub-neg 27.8%

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

      6. *-commutative 27.8%

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

      7. add-exp-log 27.8%

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

      8. expm1-define 27.8%

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

      9. log-pow 41.3%

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

      10. log1p-define 98.6%

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

   6. Applied egg-rr 98.6%

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

   if 5.00000000000000018e-285 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0 77.1%

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

      1. *-commutative 77.1%

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

   5. Simplified 77.1%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-46}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.05 \cdot 10^{+121} \lor \neg \left(i \leq 8.5 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.05e+121) (not (<= i 8.5e+141)))
   (* (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) 100.0)
   (* (* n 100.0) (/ (expm1 i) i))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.05e+121) || !(i <= 8.5e+141)) {
		tmp = ((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0;
	} else {
		tmp = (n * 100.0) * (expm1(i) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.05e+121) || !(i <= 8.5e+141)) {
		tmp = ((Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0;
	} else {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.05e+121) or not (i <= 8.5e+141):
		tmp = ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0
	else:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.05e+121) || !(i <= 8.5e+141))
		tmp = Float64(Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) * 100.0);
	else
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.05e+121], N[Not[LessEqual[i, 8.5e+141]], $MachinePrecision]], N[(N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.0500000000000001e121 or 8.4999999999999996e141 < i

   1. Initial program 71.8%

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

   if -1.0500000000000001e121 < i < 8.4999999999999996e141

   1. Initial program 17.5%

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

      1. associate-/r/ 17.8%

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

      2. associate-*r* 17.8%

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

      3. *-commutative 17.8%

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

      4. associate-*r/ 17.8%

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

      5. sub-neg 17.8%

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

      6. distribute-lft-in 17.8%

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

      7. metadata-eval 17.8%

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

      8. metadata-eval 17.8%

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

      9. metadata-eval 17.8%

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

      10. fma-define 17.8%

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

      11. metadata-eval 17.8%

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

   3. Simplified 17.8%

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

   5. Taylor expanded in n around inf 26.1%

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

      1. sub-neg 26.1%

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

      2. metadata-eval 26.1%

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

      3. metadata-eval 26.1%

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

      4. distribute-lft-in 26.1%

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

      5. metadata-eval 26.1%

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

      6. sub-neg 26.1%

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

      7. expm1-define 74.4%

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

   7. Simplified 74.4%

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

      1. associate-*r* 74.6%

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

      2. *-un-lft-identity 74.6%

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

      3. times-frac 77.5%

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

   9. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{1} \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.05 \cdot 10^{+121} \lor \neg \left(i \leq 8.5 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{+120} \lor \neg \left(i \leq 1.8 \cdot 10^{+138}\right):\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -3.5e+120) (not (<= i 1.8e+138)))
   (* n (/ (+ (* (pow (+ 1.0 (/ i n)) n) 100.0) -100.0) i))
   (* (* n 100.0) (/ (expm1 i) i))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -3.5e+120) || !(i <= 1.8e+138)) {
		tmp = n * (((pow((1.0 + (i / n)), n) * 100.0) + -100.0) / i);
	} else {
		tmp = (n * 100.0) * (expm1(i) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -3.5e+120) || !(i <= 1.8e+138)) {
		tmp = n * (((Math.pow((1.0 + (i / n)), n) * 100.0) + -100.0) / i);
	} else {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -3.5e+120) or not (i <= 1.8e+138):
		tmp = n * (((math.pow((1.0 + (i / n)), n) * 100.0) + -100.0) / i)
	else:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -3.5e+120) || !(i <= 1.8e+138))
		tmp = Float64(n * Float64(Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) * 100.0) + -100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -3.5e+120], N[Not[LessEqual[i, 1.8e+138]], $MachinePrecision]], N[(n * N[(N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -3.50000000000000007e120 or 1.8000000000000001e138 < i

   1. Initial program 71.8%

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

      1. associate-/r/ 71.9%

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

      2. associate-*r* 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-*r/ 72.0%

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

      5. sub-neg 72.0%

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

      6. distribute-lft-in 72.2%

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

      7. metadata-eval 72.2%

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

      8. metadata-eval 72.2%

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

      9. metadata-eval 72.2%

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

      10. fma-define 72.0%

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

      11. metadata-eval 72.0%

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

   3. Simplified 72.0%

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

      1. fma-undefine 72.2%

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

      2. *-commutative 72.2%

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

   6. Applied egg-rr 72.2%

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

   if -3.50000000000000007e120 < i < 1.8000000000000001e138

   1. Initial program 17.5%

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

      1. associate-/r/ 17.8%

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

      2. associate-*r* 17.8%

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

      3. *-commutative 17.8%

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

      4. associate-*r/ 17.8%

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

      5. sub-neg 17.8%

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

      6. distribute-lft-in 17.8%

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

      7. metadata-eval 17.8%

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

      8. metadata-eval 17.8%

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

      9. metadata-eval 17.8%

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

      10. fma-define 17.8%

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

      11. metadata-eval 17.8%

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

   3. Simplified 17.8%

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

   5. Taylor expanded in n around inf 26.1%

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

      1. sub-neg 26.1%

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

      2. metadata-eval 26.1%

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

      3. metadata-eval 26.1%

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

      4. distribute-lft-in 26.1%

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

      5. metadata-eval 26.1%

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

      6. sub-neg 26.1%

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

      7. expm1-define 74.4%

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

   7. Simplified 74.4%

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

      1. associate-*r* 74.6%

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

      2. *-un-lft-identity 74.6%

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

      3. times-frac 77.5%

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

   9. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{1} \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{+120} \lor \neg \left(i \leq 1.8 \cdot 10^{+138}\right):\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{+257}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq -5.4 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.9:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.6e+257)
   (* n 100.0)
   (if (<= n -5.4e-224)
     (* 100.0 (/ (expm1 i) (/ i n)))
     (if (<= n 5.5e-167)
       (/ 0.0 (/ i n))
       (if (<= n 1.9)
         (* 100.0 (/ i (/ i n)))
         (/
          (*
           n
           (*
            i
            (+
             100.0
             (*
              i
              (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))
          i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.6e+257) {
		tmp = n * 100.0;
	} else if (n <= -5.4e-224) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 5.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.6e+257) {
		tmp = n * 100.0;
	} else if (n <= -5.4e-224) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 5.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.6e+257:
		tmp = n * 100.0
	elif n <= -5.4e-224:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 5.5e-167:
		tmp = 0.0 / (i / n)
	elif n <= 1.9:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.6e+257)
		tmp = Float64(n * 100.0);
	elseif (n <= -5.4e-224)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 5.5e-167)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.9)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.6e+257], N[(n * 100.0), $MachinePrecision], If[LessEqual[n, -5.4e-224], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e-167], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.9], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -7.59999999999999996e257

   1. Initial program 12.7%

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

   3. Taylor expanded in i around 0 87.6%

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

      1. *-commutative 87.6%

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

   5. Simplified 87.6%

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

   if -7.59999999999999996e257 < n < -5.39999999999999996e-224

   1. Initial program 36.0%

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

   3. Taylor expanded in n around inf 29.5%

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

      1. expm1-define 60.9%

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

   5. Simplified 60.9%

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

   if -5.39999999999999996e-224 < n < 5.5000000000000003e-167

   1. Initial program 63.5%

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

      1. associate-*r/ 63.5%

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

      2. sub-neg 63.5%

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

      3. distribute-rgt-in 63.5%

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

      4. metadata-eval 63.5%

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

      5. metadata-eval 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in i around 0 82.4%

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

   if 5.5000000000000003e-167 < n < 1.8999999999999999

   1. Initial program 13.6%

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

   3. Taylor expanded in i around 0 65.9%

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

   if 1.8999999999999999 < n

   1. Initial program 21.9%

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

      1. associate-/r/ 22.4%

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

      2. associate-*r* 22.4%

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

      3. *-commutative 22.4%

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

      4. associate-*r/ 22.4%

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

      5. sub-neg 22.4%

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

      6. distribute-lft-in 22.4%

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

      7. metadata-eval 22.4%

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

      8. metadata-eval 22.4%

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

      9. metadata-eval 22.4%

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

      10. fma-define 22.4%

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

      11. metadata-eval 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in n around inf 34.5%

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

      1. sub-neg 34.5%

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

      2. metadata-eval 34.5%

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

      3. metadata-eval 34.5%

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

      4. distribute-lft-in 34.5%

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

      5. metadata-eval 34.5%

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

      6. sub-neg 34.5%

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

      7. expm1-define 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in i around 0 92.2%

      \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)\right)}}{i} \]
   9. Step-by-step derivation

      1. *-commutative 92.2%

         \[\leadsto \frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right)\right)}{i} \]

   10. Simplified 92.2%

       \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}}{i} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{+257}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq -5.4 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.9:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-104}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.05:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.6e-104)
   (* 100.0 (/ 1.0 (/ i (* n (expm1 i)))))
   (if (<= n 9.5e-167)
     (/ 0.0 (/ i n))
     (if (<= n 2.05)
       (* 100.0 (/ i (/ i n)))
       (/
        (*
         n
         (*
          i
          (+
           100.0
           (*
            i
            (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))
        i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.6e-104) {
		tmp = 100.0 * (1.0 / (i / (n * expm1(i))));
	} else if (n <= 9.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.05) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.6e-104) {
		tmp = 100.0 * (1.0 / (i / (n * Math.expm1(i))));
	} else if (n <= 9.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.05) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.6e-104:
		tmp = 100.0 * (1.0 / (i / (n * math.expm1(i))))
	elif n <= 9.5e-167:
		tmp = 0.0 / (i / n)
	elif n <= 2.05:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.6e-104)
		tmp = Float64(100.0 * Float64(1.0 / Float64(i / Float64(n * expm1(i)))));
	elseif (n <= 9.5e-167)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.05)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.6e-104], N[(100.0 * N[(1.0 / N[(i / N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-167], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.05], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.5999999999999999e-104

   1. Initial program 30.6%

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

   3. Taylor expanded in n around inf 26.4%

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

      1. expm1-define 57.1%

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

   5. Simplified 57.1%

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

      1. clear-num 57.1%

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

      2. inv-pow 57.1%

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

   7. Applied egg-rr 57.1%

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

      1. unpow-1 57.1%

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

      2. associate-/l/ 71.4%

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

      3. *-commutative 71.4%

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

   9. Simplified 71.4%

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

   if -4.5999999999999999e-104 < n < 9.49999999999999955e-167

   1. Initial program 59.2%

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

      1. associate-*r/ 59.2%

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

      2. sub-neg 59.2%

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

      3. distribute-rgt-in 59.2%

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

      4. metadata-eval 59.2%

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

      5. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in i around 0 71.3%

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

   if 9.49999999999999955e-167 < n < 2.0499999999999998

   1. Initial program 13.6%

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

   3. Taylor expanded in i around 0 65.9%

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

   if 2.0499999999999998 < n

   1. Initial program 21.9%

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

      1. associate-/r/ 22.4%

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

      2. associate-*r* 22.4%

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

      3. *-commutative 22.4%

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

      4. associate-*r/ 22.4%

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

      5. sub-neg 22.4%

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

      6. distribute-lft-in 22.4%

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

      7. metadata-eval 22.4%

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

      8. metadata-eval 22.4%

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

      9. metadata-eval 22.4%

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

      10. fma-define 22.4%

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

      11. metadata-eval 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in n around inf 34.5%

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

      1. sub-neg 34.5%

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

      2. metadata-eval 34.5%

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

      3. metadata-eval 34.5%

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

      4. distribute-lft-in 34.5%

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

      5. metadata-eval 34.5%

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

      6. sub-neg 34.5%

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

      7. expm1-define 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in i around 0 92.2%

      \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)\right)}}{i} \]
   9. Step-by-step derivation

      1. *-commutative 92.2%

         \[\leadsto \frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right)\right)}{i} \]

   10. Simplified 92.2%

       \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-104}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{i}{n \cdot \mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.05:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-107}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.05:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8e-107)
   (/ (* n (* 100.0 (expm1 i))) i)
   (if (<= n 5.5e-167)
     (/ 0.0 (/ i n))
     (if (<= n 2.05)
       (* 100.0 (/ i (/ i n)))
       (/
        (*
         n
         (*
          i
          (+
           100.0
           (*
            i
            (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))
        i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8e-107) {
		tmp = (n * (100.0 * expm1(i))) / i;
	} else if (n <= 5.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.05) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8e-107) {
		tmp = (n * (100.0 * Math.expm1(i))) / i;
	} else if (n <= 5.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.05) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8e-107:
		tmp = (n * (100.0 * math.expm1(i))) / i
	elif n <= 5.5e-167:
		tmp = 0.0 / (i / n)
	elif n <= 2.05:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8e-107)
		tmp = Float64(Float64(n * Float64(100.0 * expm1(i))) / i);
	elseif (n <= 5.5e-167)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.05)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8e-107], N[(N[(n * N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 5.5e-167], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.05], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8e-107

   1. Initial program 30.6%

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

      1. associate-/r/ 30.8%

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

      2. associate-*r* 30.8%

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

      3. *-commutative 30.8%

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

      4. associate-*r/ 30.9%

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

      5. sub-neg 30.9%

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

      6. distribute-lft-in 30.9%

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

      7. metadata-eval 30.9%

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

      8. metadata-eval 30.9%

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

      9. metadata-eval 30.9%

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

      10. fma-define 30.9%

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

      11. metadata-eval 30.9%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in n around inf 26.6%

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

      1. sub-neg 26.6%

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

      2. metadata-eval 26.6%

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

      3. metadata-eval 26.6%

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

      4. distribute-lft-in 26.6%

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

      5. metadata-eval 26.6%

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

      6. sub-neg 26.6%

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

      7. expm1-define 71.2%

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

   7. Simplified 71.2%

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

   if -8e-107 < n < 5.5000000000000003e-167

   1. Initial program 59.2%

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

      1. associate-*r/ 59.2%

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

      2. sub-neg 59.2%

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

      3. distribute-rgt-in 59.2%

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

      4. metadata-eval 59.2%

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

      5. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in i around 0 71.3%

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

   if 5.5000000000000003e-167 < n < 2.0499999999999998

   1. Initial program 13.6%

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

   3. Taylor expanded in i around 0 65.9%

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

   if 2.0499999999999998 < n

   1. Initial program 21.9%

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

      1. associate-/r/ 22.4%

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

      2. associate-*r* 22.4%

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

      3. *-commutative 22.4%

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

      4. associate-*r/ 22.4%

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

      5. sub-neg 22.4%

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

      6. distribute-lft-in 22.4%

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

      7. metadata-eval 22.4%

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

      8. metadata-eval 22.4%

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

      9. metadata-eval 22.4%

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

      10. fma-define 22.4%

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

      11. metadata-eval 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in n around inf 34.5%

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

      1. sub-neg 34.5%

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

      2. metadata-eval 34.5%

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

      3. metadata-eval 34.5%

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

      4. distribute-lft-in 34.5%

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

      5. metadata-eval 34.5%

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

      6. sub-neg 34.5%

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

      7. expm1-define 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in i around 0 92.2%

      \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)\right)}}{i} \]
   9. Step-by-step derivation

      1. *-commutative 92.2%

         \[\leadsto \frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right)\right)}{i} \]

   10. Simplified 92.2%

       \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-107}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.05:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.9 \cdot 10^{-224}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.9e-224)
   (* (* n 100.0) (/ (expm1 i) i))
   (if (<= n 8.5e-163)
     (/ 0.0 (/ i n))
     (if (<= n 1.95)
       (* 100.0 (/ i (/ i n)))
       (/
        (*
         n
         (*
          i
          (+
           100.0
           (*
            i
            (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))
        i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.9e-224) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (n <= 8.5e-163) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.95) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.9e-224) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (n <= 8.5e-163) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.95) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.9e-224:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif n <= 8.5e-163:
		tmp = 0.0 / (i / n)
	elif n <= 1.95:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.9e-224)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (n <= 8.5e-163)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.95)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.9e-224], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.5e-163], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.8999999999999998e-224

   1. Initial program 34.4%

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

      1. associate-/r/ 34.6%

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

      2. associate-*r* 34.6%

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

      3. *-commutative 34.6%

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

      4. associate-*r/ 34.6%

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

      5. sub-neg 34.6%

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

      6. distribute-lft-in 34.7%

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

      7. metadata-eval 34.7%

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

      8. metadata-eval 34.7%

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

      9. metadata-eval 34.7%

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

      10. fma-define 34.6%

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

      11. metadata-eval 34.6%

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

   3. Simplified 34.6%

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

   5. Taylor expanded in n around inf 27.7%

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

      1. sub-neg 27.7%

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

      2. metadata-eval 27.7%

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

      3. metadata-eval 27.7%

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

      4. distribute-lft-in 27.7%

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

      5. metadata-eval 27.7%

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

      6. sub-neg 27.7%

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

      7. expm1-define 64.3%

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

   7. Simplified 64.3%

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

      1. associate-*r* 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. times-frac 69.3%

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

   9. Applied egg-rr 69.3%

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

   if -3.8999999999999998e-224 < n < 8.5e-163

   1. Initial program 63.5%

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

      1. associate-*r/ 63.5%

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

      2. sub-neg 63.5%

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

      3. distribute-rgt-in 63.5%

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

      4. metadata-eval 63.5%

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

      5. metadata-eval 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in i around 0 82.4%

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

   if 8.5e-163 < n < 1.94999999999999996

   1. Initial program 13.6%

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

   3. Taylor expanded in i around 0 65.9%

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

   if 1.94999999999999996 < n

   1. Initial program 21.9%

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

      1. associate-/r/ 22.4%

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

      2. associate-*r* 22.4%

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

      3. *-commutative 22.4%

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

      4. associate-*r/ 22.4%

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

      5. sub-neg 22.4%

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

      6. distribute-lft-in 22.4%

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

      7. metadata-eval 22.4%

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

      8. metadata-eval 22.4%

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

      9. metadata-eval 22.4%

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

      10. fma-define 22.4%

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

      11. metadata-eval 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in n around inf 34.5%

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

      1. sub-neg 34.5%

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

      2. metadata-eval 34.5%

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

      3. metadata-eval 34.5%

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

      4. distribute-lft-in 34.5%

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

      5. metadata-eval 34.5%

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

      6. sub-neg 34.5%

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

      7. expm1-define 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in i around 0 92.2%

      \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)\right)}}{i} \]
   9. Step-by-step derivation

      1. *-commutative 92.2%

         \[\leadsto \frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right)\right)}{i} \]

   10. Simplified 92.2%

       \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.9 \cdot 10^{-224}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i \cdot \left(n \cdot \left(i \cdot 50\right)\right)}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-270}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-281}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 31500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 2.95 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* i (* n (* i 50.0))) i)) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -8e+45)
     (* (* i n) (/ 100.0 i))
     (if (<= n -1e-270)
       t_1
       (if (<= n 4.5e-281)
         t_0
         (if (<= n 31500000.0)
           t_1
           (if (<= n 2.95e+41) t_0 (+ (* n 100.0) (* 50.0 (* i n))))))))))

C

double code(double i, double n) {
	double t_0 = (i * (n * (i * 50.0))) / i;
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -8e+45) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -1e-270) {
		tmp = t_1;
	} else if (n <= 4.5e-281) {
		tmp = t_0;
	} else if (n <= 31500000.0) {
		tmp = t_1;
	} else if (n <= 2.95e+41) {
		tmp = t_0;
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (i * (n * (i * 50.0d0))) / i
    t_1 = 100.0d0 * (i / (i / n))
    if (n <= (-8d+45)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= (-1d-270)) then
        tmp = t_1
    else if (n <= 4.5d-281) then
        tmp = t_0
    else if (n <= 31500000.0d0) then
        tmp = t_1
    else if (n <= 2.95d+41) then
        tmp = t_0
    else
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (i * (n * (i * 50.0))) / i;
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -8e+45) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -1e-270) {
		tmp = t_1;
	} else if (n <= 4.5e-281) {
		tmp = t_0;
	} else if (n <= 31500000.0) {
		tmp = t_1;
	} else if (n <= 2.95e+41) {
		tmp = t_0;
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i * (n * (i * 50.0))) / i
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -8e+45:
		tmp = (i * n) * (100.0 / i)
	elif n <= -1e-270:
		tmp = t_1
	elif n <= 4.5e-281:
		tmp = t_0
	elif n <= 31500000.0:
		tmp = t_1
	elif n <= 2.95e+41:
		tmp = t_0
	else:
		tmp = (n * 100.0) + (50.0 * (i * n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i * Float64(n * Float64(i * 50.0))) / i)
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -8e+45)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= -1e-270)
		tmp = t_1;
	elseif (n <= 4.5e-281)
		tmp = t_0;
	elseif (n <= 31500000.0)
		tmp = t_1;
	elseif (n <= 2.95e+41)
		tmp = t_0;
	else
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (i * (n * (i * 50.0))) / i;
	t_1 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -8e+45)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= -1e-270)
		tmp = t_1;
	elseif (n <= 4.5e-281)
		tmp = t_0;
	elseif (n <= 31500000.0)
		tmp = t_1;
	elseif (n <= 2.95e+41)
		tmp = t_0;
	else
		tmp = (n * 100.0) + (50.0 * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i * N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -8e+45], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1e-270], t$95$1, If[LessEqual[n, 4.5e-281], t$95$0, If[LessEqual[n, 31500000.0], t$95$1, If[LessEqual[n, 2.95e+41], t$95$0, N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -7.9999999999999994e45

   1. Initial program 30.2%

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

      1. associate-/r/ 30.6%

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

      2. associate-*r* 30.5%

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

      3. *-commutative 30.5%

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

      4. associate-*r/ 30.7%

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

      5. sub-neg 30.7%

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

      6. distribute-lft-in 30.7%

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

      7. metadata-eval 30.7%

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

      8. metadata-eval 30.7%

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

      9. metadata-eval 30.7%

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

      10. fma-define 30.7%

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

      11. metadata-eval 30.7%

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

   3. Simplified 30.7%

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

   5. Taylor expanded in n around inf 33.4%

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

      1. sub-neg 33.4%

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

      2. metadata-eval 33.4%

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

      3. metadata-eval 33.4%

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

      4. distribute-lft-in 33.4%

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

      5. metadata-eval 33.4%

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

      6. sub-neg 33.4%

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

      7. expm1-define 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in i around 0 52.0%

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

      1. *-commutative 52.0%

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

   10. Simplified 52.0%

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

       1. *-commutative 52.0%

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

       2. associate-/l* 52.2%

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

       3. *-commutative 52.2%

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

   12. Applied egg-rr 52.2%

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

   if -7.9999999999999994e45 < n < -1e-270 or 4.49999999999999993e-281 < n < 3.15e7

   1. Initial program 36.2%

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

   3. Taylor expanded in i around 0 54.0%

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

   if -1e-270 < n < 4.49999999999999993e-281 or 3.15e7 < n < 2.95e41

   1. Initial program 82.2%

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

      1. associate-/r/ 82.3%

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

      2. associate-*r* 82.3%

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

      3. *-commutative 82.3%

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

      4. associate-*r/ 82.3%

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

      5. sub-neg 82.3%

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

      6. distribute-lft-in 82.3%

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

      7. metadata-eval 82.3%

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

      8. metadata-eval 82.3%

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

      9. metadata-eval 82.3%

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

      10. fma-define 82.3%

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

      11. metadata-eval 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in n around inf 75.0%

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

      1. sub-neg 75.0%

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

      2. metadata-eval 75.0%

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

      3. metadata-eval 75.0%

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

      4. distribute-lft-in 74.9%

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

      5. metadata-eval 74.9%

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

      6. sub-neg 74.9%

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

      7. expm1-define 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in i around 0 74.3%

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

      1. +-commutative 74.3%

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

      2. associate-*r* 74.3%

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

      3. distribute-rgt-in 74.3%

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

      4. *-commutative 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in i around inf 69.9%

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

   if 2.95e41 < n

   1. Initial program 13.3%

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

      1. associate-/r/ 13.9%

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

      2. associate-*r* 13.9%

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

      3. *-commutative 13.9%

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

      4. associate-*r/ 13.9%

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

      5. sub-neg 13.9%

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

      6. distribute-lft-in 13.9%

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

      7. metadata-eval 13.9%

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

      8. metadata-eval 13.9%

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

      9. metadata-eval 13.9%

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

      10. fma-define 13.9%

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

      11. metadata-eval 13.9%

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

   3. Simplified 13.9%

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

   5. Taylor expanded in n around inf 30.4%

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

      1. sub-neg 30.4%

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

      2. metadata-eval 30.4%

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

      3. metadata-eval 30.4%

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

      4. distribute-lft-in 30.5%

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

      5. metadata-eval 30.5%

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

      6. sub-neg 30.5%

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

      7. expm1-define 95.2%

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

   7. Simplified 95.2%

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

   8. Taylor expanded in i around 0 84.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-270}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;n \leq 31500000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.95 \cdot 10^{+41}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(i \cdot 50\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-166}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (/
          (*
           n
           (*
            i
            (+
             100.0
             (*
              i
              (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))
          i)))
   (if (<= n -1.12e-33)
     t_0
     (if (<= n 6e-166)
       (/ 0.0 (/ i n))
       (if (<= n 1.8) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	double tmp;
	if (n <= -1.12e-33) {
		tmp = t_0;
	} else if (n <= 6e-166) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (n * (i * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0)))))))) / i
    if (n <= (-1.12d-33)) then
        tmp = t_0
    else if (n <= 6d-166) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 1.8d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	double tmp;
	if (n <= -1.12e-33) {
		tmp = t_0;
	} else if (n <= 6e-166) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i
	tmp = 0
	if n <= -1.12e-33:
		tmp = t_0
	elif n <= 6e-166:
		tmp = 0.0 / (i / n)
	elif n <= 1.8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))) / i)
	tmp = 0.0
	if (n <= -1.12e-33)
		tmp = t_0;
	elseif (n <= 6e-166)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	tmp = 0.0;
	if (n <= -1.12e-33)
		tmp = t_0;
	elseif (n <= 6e-166)
		tmp = 0.0 / (i / n);
	elseif (n <= 1.8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.12e-33], t$95$0, If[LessEqual[n, 6e-166], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.11999999999999999e-33 or 1.80000000000000004 < n

   1. Initial program 26.7%

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

      1. associate-/r/ 27.1%

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

      2. associate-*r* 27.1%

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

      3. *-commutative 27.1%

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

      4. associate-*r/ 27.1%

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

      5. sub-neg 27.1%

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

      6. distribute-lft-in 27.1%

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

      7. metadata-eval 27.1%

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

      8. metadata-eval 27.1%

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

      9. metadata-eval 27.1%

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

      10. fma-define 27.1%

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

      11. metadata-eval 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in n around inf 32.0%

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

      1. sub-neg 32.0%

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

      2. metadata-eval 32.0%

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

      3. metadata-eval 32.0%

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

      4. distribute-lft-in 32.0%

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

      5. metadata-eval 32.0%

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

      6. sub-neg 32.0%

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

      7. expm1-define 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in i around 0 74.8%

      \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)\right)}}{i} \]
   9. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto \frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right)\right)}{i} \]

   10. Simplified 74.8%

       \[\leadsto \frac{n \cdot \color{blue}{\left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}}{i} \]

   if -1.11999999999999999e-33 < n < 6.0000000000000005e-166

   1. Initial program 53.3%

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

      1. associate-*r/ 53.3%

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

      2. sub-neg 53.3%

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

      3. distribute-rgt-in 53.3%

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

      4. metadata-eval 53.3%

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

      5. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in i around 0 63.2%

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

   if 6.0000000000000005e-166 < n < 1.80000000000000004

   1. Initial program 13.6%

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

   3. Taylor expanded in i around 0 65.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-166}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(16.666666666666668 \cdot \left(i \cdot n\right) + n \cdot 50\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-166}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)}{\frac{1}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-33)
   (+ (* n 100.0) (* i (+ (* 16.666666666666668 (* i n)) (* n 50.0))))
   (if (<= n 2.1e-166)
     (/ 0.0 (/ i n))
     (if (<= n 1.95)
       (* 100.0 (/ i (/ i n)))
       (/
        (+
         100.0
         (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))
        (/ 1.0 n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)));
	} else if (n <= 2.1e-166) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.95) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))) / (1.0 / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.12d-33)) then
        tmp = (n * 100.0d0) + (i * ((16.666666666666668d0 * (i * n)) + (n * 50.0d0)))
    else if (n <= 2.1d-166) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 1.95d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0)))))) / (1.0d0 / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)));
	} else if (n <= 2.1e-166) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.95) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))) / (1.0 / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-33:
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)))
	elif n <= 2.1e-166:
		tmp = 0.0 / (i / n)
	elif n <= 1.95:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))) / (1.0 / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-33)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(16.666666666666668 * Float64(i * n)) + Float64(n * 50.0))));
	elseif (n <= 2.1e-166)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.95)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))) / Float64(1.0 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.12e-33)
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)));
	elseif (n <= 2.1e-166)
		tmp = 0.0 / (i / n);
	elseif (n <= 1.95)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))) / (1.0 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-33], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(16.666666666666668 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-166], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.11999999999999999e-33

   1. Initial program 31.2%

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

      1. associate-/r/ 31.5%

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

      2. associate-*r* 31.4%

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

      3. *-commutative 31.4%

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

      4. associate-*r/ 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-lft-in 31.6%

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

      7. metadata-eval 31.6%

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

      8. metadata-eval 31.6%

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

      9. metadata-eval 31.6%

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

      10. fma-define 31.5%

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

      11. metadata-eval 31.5%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in n around inf 29.7%

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

      1. sub-neg 29.7%

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

      2. metadata-eval 29.7%

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

      3. metadata-eval 29.7%

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

      4. distribute-lft-in 29.7%

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

      5. metadata-eval 29.7%

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

      6. sub-neg 29.7%

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

      7. expm1-define 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in i around 0 55.8%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(16.666666666666668 \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]

   if -1.11999999999999999e-33 < n < 2.0999999999999999e-166

   1. Initial program 53.3%

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

      1. associate-*r/ 53.3%

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

      2. sub-neg 53.3%

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

      3. distribute-rgt-in 53.3%

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

      4. metadata-eval 53.3%

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

      5. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in i around 0 63.2%

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

   if 2.0999999999999999e-166 < n < 1.94999999999999996

   1. Initial program 13.6%

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

   3. Taylor expanded in i around 0 65.9%

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

   if 1.94999999999999996 < n

   1. Initial program 21.9%

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

   3. Taylor expanded in n around inf 34.0%

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

      1. expm1-define 67.5%

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

   5. Simplified 67.5%

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

      1. associate-*r/ 67.3%

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

      2. div-inv 67.3%

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

      3. associate-/r* 92.0%

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

   7. Applied egg-rr 92.0%

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

   8. Taylor expanded in i around 0 91.2%

      \[\leadsto \frac{\color{blue}{100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)}}{\frac{1}{n}} \]
   9. Step-by-step derivation

      1. *-commutative 91.2%

         \[\leadsto \frac{100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)}{\frac{1}{n}} \]

   10. Simplified 91.2%

       \[\leadsto \frac{\color{blue}{100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)}}{\frac{1}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(16.666666666666668 \cdot \left(i \cdot n\right) + n \cdot 50\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-166}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)}{\frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-108}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 2.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 70000000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(i \cdot 50\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.6e-108)
   (* (* i n) (/ 100.0 i))
   (if (<= n 2.75e-163)
     (/ 0.0 (/ i n))
     (if (<= n 70000000000.0)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 2.6e+42)
         (/ (* i (* n (* i 50.0))) i)
         (+ (* n 100.0) (* 50.0 (* i n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.6e-108) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 2.75e-163) {
		tmp = 0.0 / (i / n);
	} else if (n <= 70000000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.6e+42) {
		tmp = (i * (n * (i * 50.0))) / i;
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.6d-108)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= 2.75d-163) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 70000000000.0d0) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 2.6d+42) then
        tmp = (i * (n * (i * 50.0d0))) / i
    else
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.6e-108) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 2.75e-163) {
		tmp = 0.0 / (i / n);
	} else if (n <= 70000000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.6e+42) {
		tmp = (i * (n * (i * 50.0))) / i;
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.6e-108:
		tmp = (i * n) * (100.0 / i)
	elif n <= 2.75e-163:
		tmp = 0.0 / (i / n)
	elif n <= 70000000000.0:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.6e+42:
		tmp = (i * (n * (i * 50.0))) / i
	else:
		tmp = (n * 100.0) + (50.0 * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.6e-108)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= 2.75e-163)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 70000000000.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.6e+42)
		tmp = Float64(Float64(i * Float64(n * Float64(i * 50.0))) / i);
	else
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.6e-108)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= 2.75e-163)
		tmp = 0.0 / (i / n);
	elseif (n <= 70000000000.0)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 2.6e+42)
		tmp = (i * (n * (i * 50.0))) / i;
	else
		tmp = (n * 100.0) + (50.0 * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.6e-108], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.75e-163], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 70000000000.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e+42], N[(N[(i * N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -2.59999999999999984e-108

   1. Initial program 30.6%

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

      1. associate-/r/ 30.8%

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

      2. associate-*r* 30.8%

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

      3. *-commutative 30.8%

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

      4. associate-*r/ 30.9%

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

      5. sub-neg 30.9%

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

      6. distribute-lft-in 30.9%

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

      7. metadata-eval 30.9%

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

      8. metadata-eval 30.9%

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

      9. metadata-eval 30.9%

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

      10. fma-define 30.9%

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

      11. metadata-eval 30.9%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in n around inf 26.6%

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

      1. sub-neg 26.6%

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

      2. metadata-eval 26.6%

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

      3. metadata-eval 26.6%

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

      4. distribute-lft-in 26.6%

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

      5. metadata-eval 26.6%

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

      6. sub-neg 26.6%

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

      7. expm1-define 71.2%

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

   7. Simplified 71.2%

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

   8. Taylor expanded in i around 0 50.8%

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

      1. *-commutative 50.8%

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

   10. Simplified 50.8%

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

       1. *-commutative 50.8%

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

       2. associate-/l* 51.0%

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

       3. *-commutative 51.0%

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

   12. Applied egg-rr 51.0%

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

   if -2.59999999999999984e-108 < n < 2.7499999999999999e-163

   1. Initial program 59.2%

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

      1. associate-*r/ 59.2%

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

      2. sub-neg 59.2%

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

      3. distribute-rgt-in 59.2%

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

      4. metadata-eval 59.2%

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

      5. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in i around 0 71.3%

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

   if 2.7499999999999999e-163 < n < 7e10

   1. Initial program 13.6%

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

   3. Taylor expanded in i around 0 65.9%

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

   if 7e10 < n < 2.5999999999999999e42

   1. Initial program 73.5%

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

      1. associate-/r/ 73.5%

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

      2. associate-*r* 73.5%

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

      3. *-commutative 73.5%

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

      4. associate-*r/ 73.5%

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

      5. sub-neg 73.5%

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

      6. distribute-lft-in 73.5%

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

      7. metadata-eval 73.5%

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

      8. metadata-eval 73.5%

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

      9. metadata-eval 73.5%

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

      10. fma-define 73.5%

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

      11. metadata-eval 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in n around inf 58.9%

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

      1. sub-neg 58.9%

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

      2. metadata-eval 58.9%

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

      3. metadata-eval 58.9%

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

      4. distribute-lft-in 58.8%

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

      5. metadata-eval 58.8%

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

      6. sub-neg 58.8%

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

      7. expm1-define 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in i around 0 82.7%

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

      1. +-commutative 82.7%

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

      2. associate-*r* 82.7%

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

      3. distribute-rgt-in 82.7%

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

      4. *-commutative 82.7%

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

   10. Simplified 82.7%

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

   11. Taylor expanded in i around inf 65.8%

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

   if 2.5999999999999999e42 < n

   1. Initial program 13.3%

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

      1. associate-/r/ 13.9%

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

      2. associate-*r* 13.9%

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

      3. *-commutative 13.9%

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

      4. associate-*r/ 13.9%

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

      5. sub-neg 13.9%

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

      6. distribute-lft-in 13.9%

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

      7. metadata-eval 13.9%

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

      8. metadata-eval 13.9%

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

      9. metadata-eval 13.9%

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

      10. fma-define 13.9%

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

      11. metadata-eval 13.9%

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

   3. Simplified 13.9%

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

   5. Taylor expanded in n around inf 30.4%

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

      1. sub-neg 30.4%

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

      2. metadata-eval 30.4%

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

      3. metadata-eval 30.4%

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

      4. distribute-lft-in 30.5%

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

      5. metadata-eval 30.5%

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

      6. sub-neg 30.5%

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

      7. expm1-define 95.2%

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

   7. Simplified 95.2%

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

   8. Taylor expanded in i around 0 84.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-108}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 2.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 70000000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(i \cdot 50\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-33)
   (/ (* i (* n (+ 100.0 (* i 50.0)))) i)
   (if (<= n 7e-167)
     (/ 0.0 (/ i n))
     (if (<= n 5.5e-8)
       (* 100.0 (/ i (/ i n)))
       (/ (* i (+ (* n 100.0) (* 50.0 (* i n)))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	} else if (n <= 7e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 5.5e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * ((n * 100.0) + (50.0 * (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 <= (-1.12d-33)) then
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    else if (n <= 7d-167) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 5.5d-8) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * ((n * 100.0d0) + (50.0d0 * (i * n)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	} else if (n <= 7e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 5.5e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-33:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	elif n <= 7e-167:
		tmp = 0.0 / (i / n)
	elif n <= 5.5e-8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-33)
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	elseif (n <= 7e-167)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 5.5e-8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.12e-33)
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	elseif (n <= 7e-167)
		tmp = 0.0 / (i / n);
	elseif (n <= 5.5e-8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-33], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 7e-167], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e-8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.11999999999999999e-33

   1. Initial program 31.2%

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

      1. associate-/r/ 31.5%

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

      2. associate-*r* 31.4%

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

      3. *-commutative 31.4%

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

      4. associate-*r/ 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-lft-in 31.6%

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

      7. metadata-eval 31.6%

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

      8. metadata-eval 31.6%

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

      9. metadata-eval 31.6%

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

      10. fma-define 31.5%

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

      11. metadata-eval 31.5%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in n around inf 29.7%

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

      1. sub-neg 29.7%

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

      2. metadata-eval 29.7%

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

      3. metadata-eval 29.7%

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

      4. distribute-lft-in 29.7%

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

      5. metadata-eval 29.7%

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

      6. sub-neg 29.7%

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

      7. expm1-define 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in i around 0 55.1%

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

      1. +-commutative 55.1%

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

      2. associate-*r* 55.1%

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

      3. distribute-rgt-in 55.1%

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

      4. *-commutative 55.1%

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

   10. Simplified 55.1%

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

   if -1.11999999999999999e-33 < n < 6.9999999999999998e-167

   1. Initial program 53.3%

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

      1. associate-*r/ 53.3%

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

      2. sub-neg 53.3%

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

      3. distribute-rgt-in 53.3%

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

      4. metadata-eval 53.3%

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

      5. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in i around 0 63.2%

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

   if 6.9999999999999998e-167 < n < 5.5000000000000003e-8

   1. Initial program 10.5%

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

   3. Taylor expanded in i around 0 67.0%

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

   if 5.5000000000000003e-8 < n

   1. Initial program 22.6%

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

      1. associate-/r/ 23.1%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.1%

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

      7. metadata-eval 23.1%

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

      8. metadata-eval 23.1%

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

      9. metadata-eval 23.1%

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

      10. fma-define 23.1%

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

      11. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in n around inf 33.7%

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

      1. sub-neg 33.7%

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

      2. metadata-eval 33.7%

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

      3. metadata-eval 33.7%

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

      4. distribute-lft-in 33.7%

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

      5. metadata-eval 33.7%

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

      6. sub-neg 33.7%

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

      7. expm1-define 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in i around 0 87.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;\frac{100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)}{\frac{1}{n}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-166}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-33)
   (/ (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))) (/ 1.0 n))
   (if (<= n 1.15e-166)
     (/ 0.0 (/ i n))
     (if (<= n 1.8e-7)
       (* 100.0 (/ i (/ i n)))
       (/ (* i (+ (* n 100.0) (* 50.0 (* i n)))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (100.0 + (i * (50.0 + (i * 16.666666666666668)))) / (1.0 / n);
	} else if (n <= 1.15e-166) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * ((n * 100.0) + (50.0 * (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 <= (-1.12d-33)) then
        tmp = (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0)))) / (1.0d0 / n)
    else if (n <= 1.15d-166) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 1.8d-7) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * ((n * 100.0d0) + (50.0d0 * (i * n)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (100.0 + (i * (50.0 + (i * 16.666666666666668)))) / (1.0 / n);
	} else if (n <= 1.15e-166) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-33:
		tmp = (100.0 + (i * (50.0 + (i * 16.666666666666668)))) / (1.0 / n)
	elif n <= 1.15e-166:
		tmp = 0.0 / (i / n)
	elif n <= 1.8e-7:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-33)
		tmp = Float64(Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))) / Float64(1.0 / n));
	elseif (n <= 1.15e-166)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.8e-7)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.12e-33)
		tmp = (100.0 + (i * (50.0 + (i * 16.666666666666668)))) / (1.0 / n);
	elseif (n <= 1.15e-166)
		tmp = 0.0 / (i / n);
	elseif (n <= 1.8e-7)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-33], N[(N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-166], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-7], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.11999999999999999e-33

   1. Initial program 31.2%

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

   3. Taylor expanded in n around inf 29.5%

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

      1. expm1-define 61.5%

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

   5. Simplified 61.5%

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

      1. associate-*r/ 61.3%

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

      2. div-inv 61.3%

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

      3. associate-/r* 78.0%

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

   7. Applied egg-rr 78.0%

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

   8. Taylor expanded in i around 0 55.7%

      \[\leadsto \frac{\color{blue}{100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)}}{\frac{1}{n}} \]
   9. Step-by-step derivation

      1. *-commutative 55.7%

         \[\leadsto \frac{100 + i \cdot \left(50 + \color{blue}{i \cdot 16.666666666666668}\right)}{\frac{1}{n}} \]

   10. Simplified 55.7%

       \[\leadsto \frac{\color{blue}{100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)}}{\frac{1}{n}} \]

   if -1.11999999999999999e-33 < n < 1.14999999999999999e-166

   1. Initial program 53.3%

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

      1. associate-*r/ 53.3%

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

      2. sub-neg 53.3%

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

      3. distribute-rgt-in 53.3%

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

      4. metadata-eval 53.3%

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

      5. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in i around 0 63.2%

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

   if 1.14999999999999999e-166 < n < 1.79999999999999997e-7

   1. Initial program 10.5%

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

   3. Taylor expanded in i around 0 67.0%

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

   if 1.79999999999999997e-7 < n

   1. Initial program 22.6%

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

      1. associate-/r/ 23.1%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.1%

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

      7. metadata-eval 23.1%

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

      8. metadata-eval 23.1%

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

      9. metadata-eval 23.1%

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

      10. fma-define 23.1%

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

      11. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in n around inf 33.7%

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

      1. sub-neg 33.7%

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

      2. metadata-eval 33.7%

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

      3. metadata-eval 33.7%

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

      4. distribute-lft-in 33.7%

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

      5. metadata-eval 33.7%

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

      6. sub-neg 33.7%

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

      7. expm1-define 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in i around 0 87.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;\frac{100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)}{\frac{1}{n}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-166}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(16.666666666666668 \cdot \left(i \cdot n\right) + n \cdot 50\right)\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-33)
   (+ (* n 100.0) (* i (+ (* 16.666666666666668 (* i n)) (* n 50.0))))
   (if (<= n 5.5e-167)
     (/ 0.0 (/ i n))
     (if (<= n 1.8e-7)
       (* 100.0 (/ i (/ i n)))
       (/ (* i (+ (* n 100.0) (* 50.0 (* i n)))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)));
	} else if (n <= 5.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * ((n * 100.0) + (50.0 * (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 <= (-1.12d-33)) then
        tmp = (n * 100.0d0) + (i * ((16.666666666666668d0 * (i * n)) + (n * 50.0d0)))
    else if (n <= 5.5d-167) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 1.8d-7) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * ((n * 100.0d0) + (50.0d0 * (i * n)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-33) {
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)));
	} else if (n <= 5.5e-167) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-33:
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)))
	elif n <= 5.5e-167:
		tmp = 0.0 / (i / n)
	elif n <= 1.8e-7:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-33)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(16.666666666666668 * Float64(i * n)) + Float64(n * 50.0))));
	elseif (n <= 5.5e-167)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.8e-7)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.12e-33)
		tmp = (n * 100.0) + (i * ((16.666666666666668 * (i * n)) + (n * 50.0)));
	elseif (n <= 5.5e-167)
		tmp = 0.0 / (i / n);
	elseif (n <= 1.8e-7)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-33], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(16.666666666666668 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e-167], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-7], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.11999999999999999e-33

   1. Initial program 31.2%

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

      1. associate-/r/ 31.5%

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

      2. associate-*r* 31.4%

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

      3. *-commutative 31.4%

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

      4. associate-*r/ 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-lft-in 31.6%

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

      7. metadata-eval 31.6%

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

      8. metadata-eval 31.6%

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

      9. metadata-eval 31.6%

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

      10. fma-define 31.5%

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

      11. metadata-eval 31.5%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in n around inf 29.7%

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

      1. sub-neg 29.7%

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

      2. metadata-eval 29.7%

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

      3. metadata-eval 29.7%

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

      4. distribute-lft-in 29.7%

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

      5. metadata-eval 29.7%

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

      6. sub-neg 29.7%

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

      7. expm1-define 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in i around 0 55.8%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(16.666666666666668 \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]

   if -1.11999999999999999e-33 < n < 5.5000000000000003e-167

   1. Initial program 53.3%

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

      1. associate-*r/ 53.3%

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

      2. sub-neg 53.3%

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

      3. distribute-rgt-in 53.3%

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

      4. metadata-eval 53.3%

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

      5. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in i around 0 63.2%

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

   if 5.5000000000000003e-167 < n < 1.79999999999999997e-7

   1. Initial program 10.5%

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

   3. Taylor expanded in i around 0 67.0%

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

   if 1.79999999999999997e-7 < n

   1. Initial program 22.6%

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

      1. associate-/r/ 23.1%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.1%

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

      7. metadata-eval 23.1%

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

      8. metadata-eval 23.1%

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

      9. metadata-eval 23.1%

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

      10. fma-define 23.1%

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

      11. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in n around inf 33.7%

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

      1. sub-neg 33.7%

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

      2. metadata-eval 33.7%

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

      3. metadata-eval 33.7%

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

      4. distribute-lft-in 33.7%

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

      5. metadata-eval 33.7%

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

      6. sub-neg 33.7%

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

      7. expm1-define 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in i around 0 87.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(16.666666666666668 \cdot \left(i \cdot n\right) + n \cdot 50\right)\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-162}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* i (* n (+ 100.0 (* i 50.0)))) i)))
   (if (<= n -1.12e-33)
     t_0
     (if (<= n 2.05e-162)
       (/ 0.0 (/ i n))
       (if (<= n 1.65e-7) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -1.12e-33) {
		tmp = t_0;
	} else if (n <= 2.05e-162) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.65e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    if (n <= (-1.12d-33)) then
        tmp = t_0
    else if (n <= 2.05d-162) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 1.65d-7) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -1.12e-33) {
		tmp = t_0;
	} else if (n <= 2.05e-162) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.65e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i
	tmp = 0
	if n <= -1.12e-33:
		tmp = t_0
	elif n <= 2.05e-162:
		tmp = 0.0 / (i / n)
	elif n <= 1.65e-7:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i)
	tmp = 0.0
	if (n <= -1.12e-33)
		tmp = t_0;
	elseif (n <= 2.05e-162)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.65e-7)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	tmp = 0.0;
	if (n <= -1.12e-33)
		tmp = t_0;
	elseif (n <= 2.05e-162)
		tmp = 0.0 / (i / n);
	elseif (n <= 1.65e-7)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.12e-33], t$95$0, If[LessEqual[n, 2.05e-162], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e-7], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.11999999999999999e-33 or 1.6500000000000001e-7 < n

   1. Initial program 27.0%

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

      1. associate-/r/ 27.4%

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

      2. associate-*r* 27.4%

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

      3. *-commutative 27.4%

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

      4. associate-*r/ 27.4%

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

      5. sub-neg 27.4%

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

      6. distribute-lft-in 27.4%

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

      7. metadata-eval 27.4%

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

      8. metadata-eval 27.4%

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

      9. metadata-eval 27.4%

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

      10. fma-define 27.4%

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

      11. metadata-eval 27.4%

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

   3. Simplified 27.4%

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

   5. Taylor expanded in n around inf 31.7%

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

      1. sub-neg 31.7%

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

      2. metadata-eval 31.7%

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

      3. metadata-eval 31.7%

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

      4. distribute-lft-in 31.7%

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

      5. metadata-eval 31.7%

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

      6. sub-neg 31.7%

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

      7. expm1-define 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in i around 0 71.2%

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

      1. +-commutative 71.2%

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

      2. associate-*r* 71.2%

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

      3. distribute-rgt-in 71.1%

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

      4. *-commutative 71.1%

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

   10. Simplified 71.1%

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

   if -1.11999999999999999e-33 < n < 2.0500000000000001e-162

   1. Initial program 53.3%

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

      1. associate-*r/ 53.3%

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

      2. sub-neg 53.3%

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

      3. distribute-rgt-in 53.3%

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

      4. metadata-eval 53.3%

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

      5. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in i around 0 63.2%

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

   if 2.0500000000000001e-162 < n < 1.6500000000000001e-7

   1. Initial program 10.5%

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

   3. Taylor expanded in i around 0 67.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-162}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.4 \cdot 10^{+45}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8.4e+45)
   (* (* i n) (/ 100.0 i))
   (if (<= n 1.8e-7)
     (* 100.0 (/ i (/ i n)))
     (+ (* n 100.0) (* 50.0 (* i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8.4e+45) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-8.4d+45)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= 1.8d-7) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8.4e+45) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8.4e+45:
		tmp = (i * n) * (100.0 / i)
	elif n <= 1.8e-7:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) + (50.0 * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8.4e+45)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= 1.8e-7)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -8.4e+45)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= 1.8e-7)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) + (50.0 * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8.4e+45], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-7], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.39999999999999979e45

   1. Initial program 30.2%

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

      1. associate-/r/ 30.6%

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

      2. associate-*r* 30.5%

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

      3. *-commutative 30.5%

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

      4. associate-*r/ 30.7%

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

      5. sub-neg 30.7%

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

      6. distribute-lft-in 30.7%

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

      7. metadata-eval 30.7%

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

      8. metadata-eval 30.7%

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

      9. metadata-eval 30.7%

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

      10. fma-define 30.7%

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

      11. metadata-eval 30.7%

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

   3. Simplified 30.7%

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

   5. Taylor expanded in n around inf 33.4%

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

      1. sub-neg 33.4%

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

      2. metadata-eval 33.4%

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

      3. metadata-eval 33.4%

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

      4. distribute-lft-in 33.4%

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

      5. metadata-eval 33.4%

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

      6. sub-neg 33.4%

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

      7. expm1-define 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in i around 0 52.0%

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

      1. *-commutative 52.0%

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

   10. Simplified 52.0%

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

       1. *-commutative 52.0%

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

       2. associate-/l* 52.2%

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

       3. *-commutative 52.2%

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

   12. Applied egg-rr 52.2%

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

   if -8.39999999999999979e45 < n < 1.79999999999999997e-7

   1. Initial program 41.3%

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

   3. Taylor expanded in i around 0 52.0%

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

   if 1.79999999999999997e-7 < n

   1. Initial program 22.6%

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

      1. associate-/r/ 23.1%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.1%

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

      7. metadata-eval 23.1%

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

      8. metadata-eval 23.1%

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

      9. metadata-eval 23.1%

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

      10. fma-define 23.1%

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

      11. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in n around inf 33.7%

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

      1. sub-neg 33.7%

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

      2. metadata-eval 33.7%

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

      3. metadata-eval 33.7%

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

      4. distribute-lft-in 33.7%

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

      5. metadata-eval 33.7%

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

      6. sub-neg 33.7%

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

      7. expm1-define 90.9%

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

   7. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}} \]

   8. Taylor expanded in i around 0 75.1%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.4 \cdot 10^{+45}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+61} \lor \neg \left(i \leq 10\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1e+61) (not (<= i 10.0))) (* 100.0 (/ i (/ i n))) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1e+61) || !(i <= 10.0)) {
		tmp = 100.0 * (i / (i / n));
	} 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 <= (-1d+61)) .or. (.not. (i <= 10.0d0))) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1e+61) || !(i <= 10.0)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1e+61) or not (i <= 10.0):
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1e+61) || !(i <= 10.0))
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -1e+61) || ~((i <= 10.0)))
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1e+61], N[Not[LessEqual[i, 10.0]], $MachinePrecision]], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -9.99999999999999949e60 or 10 < i

   1. Initial program 57.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 22.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if -9.99999999999999949e60 < i < 10

   1. Initial program 13.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 79.3%

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-commutative 79.3%

         \[\leadsto \color{blue}{n \cdot 100} \]

   5. Simplified 79.3%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+61} \lor \neg \left(i \leq 10\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+45} \lor \neg \left(n \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -8e+45) (not (<= n 1.8e-7)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -8e+45) || !(n <= 1.8e-7)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8d+45)) .or. (.not. (n <= 1.8d-7))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -8e+45) || !(n <= 1.8e-7)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -8e+45) or not (n <= 1.8e-7):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -8e+45) || !(n <= 1.8e-7))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -8e+45) || ~((n <= 1.8e-7)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -8e+45], N[Not[LessEqual[n, 1.8e-7]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.9999999999999994e45 or 1.79999999999999997e-7 < n

   1. Initial program 26.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 26.5%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 26.5%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 26.5%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 26.5%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 26.5%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 26.5%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. metadata-eval 26.5%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{i} \]

      8. metadata-eval 26.5%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{i} \]

      9. metadata-eval 26.5%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{i} \]

      10. fma-define 26.5%

          \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{i} \]

      11. metadata-eval 26.5%

          \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 26.5%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 68.8%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}}{i} \]
   6. Step-by-step derivation

      1. *-commutative 68.8%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + \color{blue}{\left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100}\right)}{i} \]

      2. associate-*r/ 68.8%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 100\right)}{i} \]

      3. metadata-eval 68.8%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 100\right)}{i} \]

   7. Simplified 68.8%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) \cdot 100\right)}}{i} \]

   8. Taylor expanded in n around inf 64.5%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]
   9. Step-by-step derivation

      1. *-commutative 64.5%

         \[\leadsto n \cdot \left(100 + \color{blue}{i \cdot 50}\right) \]

   10. Simplified 64.5%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot 50\right)} \]

   if -7.9999999999999994e45 < n < 1.79999999999999997e-7

   1. Initial program 41.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 52.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+45} \lor \neg \left(n \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8e+45)
   (* (* i n) (/ 100.0 i))
   (if (<= n 1.8e-7) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8e+45) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-8d+45)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= 1.8d-7) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8e+45) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= 1.8e-7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8e+45:
		tmp = (i * n) * (100.0 / i)
	elif n <= 1.8e-7:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8e+45)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= 1.8e-7)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -8e+45)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= 1.8e-7)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8e+45], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-7], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.9999999999999994e45

   1. Initial program 30.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 30.6%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 30.5%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 30.5%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 30.7%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 30.7%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 30.7%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. metadata-eval 30.7%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{i} \]

      8. metadata-eval 30.7%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{i} \]

      9. metadata-eval 30.7%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{i} \]

      10. fma-define 30.7%

          \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{i} \]

      11. metadata-eval 30.7%

          \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 30.7%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 33.4%

      \[\leadsto \color{blue}{\frac{n \cdot \left(100 \cdot e^{i} - 100\right)}{i}} \]
   6. Step-by-step derivation

      1. sub-neg 33.4%

         \[\leadsto \frac{n \cdot \color{blue}{\left(100 \cdot e^{i} + \left(-100\right)\right)}}{i} \]

      2. metadata-eval 33.4%

         \[\leadsto \frac{n \cdot \left(100 \cdot e^{i} + \color{blue}{-100}\right)}{i} \]

      3. metadata-eval 33.4%

         \[\leadsto \frac{n \cdot \left(100 \cdot e^{i} + \color{blue}{100 \cdot -1}\right)}{i} \]

      4. distribute-lft-in 33.4%

         \[\leadsto \frac{n \cdot \color{blue}{\left(100 \cdot \left(e^{i} + -1\right)\right)}}{i} \]

      5. metadata-eval 33.4%

         \[\leadsto \frac{n \cdot \left(100 \cdot \left(e^{i} + \color{blue}{\left(-1\right)}\right)\right)}{i} \]

      6. sub-neg 33.4%

         \[\leadsto \frac{n \cdot \left(100 \cdot \color{blue}{\left(e^{i} - 1\right)}\right)}{i} \]

      7. expm1-define 78.3%

         \[\leadsto \frac{n \cdot \left(100 \cdot \color{blue}{\mathsf{expm1}\left(i\right)}\right)}{i} \]

   7. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}} \]

   8. Taylor expanded in i around 0 52.0%

      \[\leadsto \frac{\color{blue}{100 \cdot \left(i \cdot n\right)}}{i} \]
   9. Step-by-step derivation

      1. *-commutative 52.0%

         \[\leadsto \frac{100 \cdot \color{blue}{\left(n \cdot i\right)}}{i} \]

   10. Simplified 52.0%

       \[\leadsto \frac{\color{blue}{100 \cdot \left(n \cdot i\right)}}{i} \]
   11. Step-by-step derivation

       1. *-commutative 52.0%

          \[\leadsto \frac{\color{blue}{\left(n \cdot i\right) \cdot 100}}{i} \]

       2. associate-/l* 52.2%

          \[\leadsto \color{blue}{\left(n \cdot i\right) \cdot \frac{100}{i}} \]

       3. *-commutative 52.2%

          \[\leadsto \color{blue}{\left(i \cdot n\right)} \cdot \frac{100}{i} \]

   12. Applied egg-rr 52.2%

       \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \frac{100}{i}} \]

   if -7.9999999999999994e45 < n < 1.79999999999999997e-7

   1. Initial program 41.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 52.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 1.79999999999999997e-7 < n

   1. Initial program 22.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 23.1%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 23.2%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 23.2%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 23.1%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 23.1%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 23.1%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. metadata-eval 23.1%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{i} \]

      8. metadata-eval 23.1%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{i} \]

      9. metadata-eval 23.1%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{i} \]

      10. fma-define 23.1%

          \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{i} \]

      11. metadata-eval 23.1%

          \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 83.0%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}}{i} \]
   6. Step-by-step derivation

      1. *-commutative 83.0%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + \color{blue}{\left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100}\right)}{i} \]

      2. associate-*r/ 83.0%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 100\right)}{i} \]

      3. metadata-eval 83.0%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 100\right)}{i} \]

   7. Simplified 83.0%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) \cdot 100\right)}}{i} \]

   8. Taylor expanded in n around inf 75.0%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]
   9. Step-by-step derivation

      1. *-commutative 75.0%

         \[\leadsto n \cdot \left(100 + \color{blue}{i \cdot 50}\right) \]

   10. Simplified 75.0%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot 50\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5.4 \cdot 10^{+54}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5.4e+54) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5.4e+54) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 5.4d+54) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 5.4e+54) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5.4e+54:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5.4e+54)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 5.4e+54)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5.4e+54], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.40000000000000022e54

   1. Initial program 28.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 57.8%

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-commutative 57.8%

         \[\leadsto \color{blue}{n \cdot 100} \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{n \cdot 100} \]

   if 5.40000000000000022e54 < i

   1. Initial program 49.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 49.6%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 49.6%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 49.6%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 49.7%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 49.7%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 49.8%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. metadata-eval 49.8%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{i} \]

      8. metadata-eval 49.8%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{i} \]

      9. metadata-eval 49.8%

         \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{i} \]

      10. fma-define 49.7%

          \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{i} \]

      11. metadata-eval 49.7%

          \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 49.7%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 39.0%

      \[\leadsto \color{blue}{\frac{n \cdot \left(100 \cdot e^{i} - 100\right)}{i}} \]
   6. Step-by-step derivation

      1. sub-neg 39.0%

         \[\leadsto \frac{n \cdot \color{blue}{\left(100 \cdot e^{i} + \left(-100\right)\right)}}{i} \]

      2. metadata-eval 39.0%

         \[\leadsto \frac{n \cdot \left(100 \cdot e^{i} + \color{blue}{-100}\right)}{i} \]

      3. metadata-eval 39.0%

         \[\leadsto \frac{n \cdot \left(100 \cdot e^{i} + \color{blue}{100 \cdot -1}\right)}{i} \]

      4. distribute-lft-in 39.0%

         \[\leadsto \frac{n \cdot \color{blue}{\left(100 \cdot \left(e^{i} + -1\right)\right)}}{i} \]

      5. metadata-eval 39.0%

         \[\leadsto \frac{n \cdot \left(100 \cdot \left(e^{i} + \color{blue}{\left(-1\right)}\right)\right)}{i} \]

      6. sub-neg 39.0%

         \[\leadsto \frac{n \cdot \left(100 \cdot \color{blue}{\left(e^{i} - 1\right)}\right)}{i} \]

      7. expm1-define 39.0%

         \[\leadsto \frac{n \cdot \left(100 \cdot \color{blue}{\mathsf{expm1}\left(i\right)}\right)}{i} \]

   7. Simplified 39.0%

      \[\leadsto \color{blue}{\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}} \]

   8. Taylor expanded in i around 0 21.0%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]

   9. Taylor expanded in i around inf 21.0%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5.4 \cdot 10^{+54}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 2.8% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 32.7%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Step-by-step derivation

   1. associate-/r/ 33.1%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

   2. associate-*r* 33.1%

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

   3. *-commutative 33.1%

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

   4. associate-*r/ 33.1%

      \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

   5. sub-neg 33.1%

      \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

   6. distribute-lft-in 33.1%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

   7. metadata-eval 33.1%

      \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{i} \]

   8. metadata-eval 33.1%

      \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{i} \]

   9. metadata-eval 33.1%

      \[\leadsto n \cdot \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{i} \]

   10. fma-define 33.1%

       \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{i} \]

   11. metadata-eval 33.1%

       \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

3. Simplified 33.1%

   \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
4. Add Preprocessing

5. Taylor expanded in i around 0 52.6%

   \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}}{i} \]
6. Step-by-step derivation

   1. *-commutative 52.6%

      \[\leadsto n \cdot \frac{i \cdot \left(100 + \color{blue}{\left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 100}\right)}{i} \]

   2. associate-*r/ 52.6%

      \[\leadsto n \cdot \frac{i \cdot \left(100 + \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 100\right)}{i} \]

   3. metadata-eval 52.6%

      \[\leadsto n \cdot \frac{i \cdot \left(100 + \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 100\right)}{i} \]

7. Simplified 52.6%

   \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right) \cdot 100\right)}}{i} \]

8. Taylor expanded in n around 0 2.5%

   \[\leadsto \color{blue}{-50 \cdot i} \]
9. Step-by-step derivation

   1. *-commutative 2.5%

      \[\leadsto \color{blue}{i \cdot -50} \]

10. Simplified 2.5%

    \[\leadsto \color{blue}{i \cdot -50} \]

11. Final simplification 2.5%

    \[\leadsto i \cdot -50 \]
12. Add Preprocessing

Alternative 22: 48.9% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * 100.0d0
end function

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 32.7%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0 46.3%

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. *-commutative 46.3%

      \[\leadsto \color{blue}{n \cdot 100} \]

5. Simplified 46.3%

   \[\leadsto \color{blue}{n \cdot 100} \]

6. Final simplification 46.3%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 34.4% 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 2024112 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))