Compound Interest

Percentage Accurate: 29.3% → 90.4%

Time: 19.7s

Alternatives: 16

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 29.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) INFINITY)
   (* n (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) i)))
   (* n 100.0)))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(n * N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.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 35.0%

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

      1. associate-*r/ 35.0%

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

      2. sub-neg 35.0%

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

      3. distribute-rgt-in 35.0%

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

      4. metadata-eval 35.0%

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

      5. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

      1. metadata-eval 35.0%

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

      2. metadata-eval 35.0%

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

      3. distribute-rgt-in 35.0%

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

      4. sub-neg 35.0%

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

      5. associate-*r/ 35.0%

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

      6. associate-/r/ 34.5%

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

      7. associate-*r* 34.5%

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

      8. add-exp-log 34.5%

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

      9. expm1-define 34.5%

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

      10. log-pow 39.0%

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

      11. log1p-define 95.1%

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

   6. Applied egg-rr 95.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0 72.7%

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

      1. *-commutative 72.7%

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

   5. Simplified 72.7%

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

4. Final simplification 91.1%

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

Alternative 2: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) INFINITY)
   (* 100.0 (* n (/ (expm1 (* n (log1p (/ i n)))) i)))
   (* n 100.0)))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(100.0 * N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.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 35.0%

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

      1. associate-/r/ 34.5%

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

      2. add-exp-log 34.5%

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

      3. expm1-define 34.5%

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

      4. log-pow 39.0%

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

      5. log1p-define 95.0%

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

   4. Applied egg-rr 95.0%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #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 72.7%

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

      1. *-commutative 72.7%

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

   5. Simplified 72.7%

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

4. Final simplification 91.1%

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

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{-245} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.8e-245) (not (<= n 3.5e-52)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.8e-245) || !(n <= 3.5e-52)) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5.8e-245) || !(n <= 3.5e-52)) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5.8e-245) or not (n <= 3.5e-52):
		tmp = n * ((100.0 * math.expm1(i)) / i)
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.8e-245) || !(n <= 3.5e-52))
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.8e-245], N[Not[LessEqual[n, 3.5e-52]], $MachinePrecision]], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if n < -5.7999999999999999e-245

   1. Initial program 24.1%

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

      1. associate-/r/ 24.4%

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

      2. associate-*r* 24.4%

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

      3. *-commutative 24.4%

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

      4. associate-*r/ 24.4%

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

      5. sub-neg 24.4%

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

      6. distribute-lft-in 24.4%

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

      7. metadata-eval 24.4%

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

      8. metadata-eval 24.4%

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

      9. metadata-eval 24.4%

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

      10. fma-define 24.4%

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

      11. metadata-eval 24.4%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in n around inf 37.2%

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

      1. sub-neg 37.2%

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

      2. metadata-eval 37.2%

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

      3. metadata-eval 37.2%

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

      4. distribute-lft-in 37.2%

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

      5. metadata-eval 37.2%

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

      6. sub-neg 37.2%

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

      7. expm1-define 84.8%

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

   7. Simplified 84.8%

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

   if -5.7999999999999999e-245 < n < 3.5e-52

   1. Initial program 50.9%

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

      1. associate-*r/ 50.9%

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

      2. sub-neg 50.9%

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

      3. distribute-rgt-in 50.9%

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

      4. metadata-eval 50.9%

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

      5. metadata-eval 50.9%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in i around 0 72.9%

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

   if 3.5e-52 < n

   1. Initial program 19.7%

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

      1. associate-*r/ 19.7%

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

      2. sub-neg 19.7%

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

      3. distribute-rgt-in 19.7%

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

      4. metadata-eval 19.7%

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

      5. metadata-eval 19.7%

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

   3. Simplified 19.7%

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

      1. metadata-eval 19.7%

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

      2. metadata-eval 19.7%

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

      3. distribute-rgt-in 19.7%

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

      4. sub-neg 19.7%

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

      5. associate-*r/ 19.7%

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

      6. associate-/r/ 20.0%

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

      7. associate-*r* 20.0%

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

      8. add-exp-log 20.0%

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

      9. expm1-define 20.0%

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

      10. log-pow 16.4%

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

      11. log1p-define 77.2%

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

   6. Applied egg-rr 77.2%

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

   7. Taylor expanded in n around inf 42.5%

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

      1. associate-*r/ 42.5%

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

      2. expm1-define 96.3%

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

   9. Simplified 96.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{-245} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-244} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2e-244) (not (<= n 3.5e-52)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2e-244) || !(n <= 3.5e-52)) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2e-244) || !(n <= 3.5e-52)) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2e-244) or not (n <= 3.5e-52):
		tmp = n * ((100.0 * math.expm1(i)) / i)
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2e-244) || !(n <= 3.5e-52))
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2e-244], N[Not[LessEqual[n, 3.5e-52]], $MachinePrecision]], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9999999999999999e-244 or 3.5e-52 < n

   1. Initial program 22.3%

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

      1. associate-/r/ 22.5%

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

      2. associate-*r* 22.5%

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

      3. *-commutative 22.5%

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

      4. associate-*r/ 22.5%

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

      5. sub-neg 22.5%

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

      6. distribute-lft-in 22.5%

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

      7. metadata-eval 22.5%

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

      8. metadata-eval 22.5%

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

      9. metadata-eval 22.5%

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

      10. fma-define 22.5%

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

      11. metadata-eval 22.5%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in n around inf 39.4%

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

      1. sub-neg 39.4%

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

      2. metadata-eval 39.4%

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

      3. metadata-eval 39.4%

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

      4. distribute-lft-in 39.4%

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

      5. metadata-eval 39.4%

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

      6. sub-neg 39.4%

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

      7. expm1-define 89.6%

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

   7. Simplified 89.6%

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

   if -1.9999999999999999e-244 < n < 3.5e-52

   1. Initial program 50.9%

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

      1. associate-*r/ 50.9%

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

      2. sub-neg 50.9%

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

      3. distribute-rgt-in 50.9%

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

      4. metadata-eval 50.9%

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

      5. metadata-eval 50.9%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in i around 0 72.9%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-244} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -4.1 \cdot 10^{-199} \lor \neg \left(i \leq 1.02 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.2e-54)
   (* 100.0 (/ (expm1 i) (/ i n)))
   (if (or (<= i -4.1e-199) (not (<= i 1.02e-162)))
     (*
      (/ 100.0 i)
      (*
       i
       (+
        n
        (*
         i
         (+
          (* n 0.5)
          (*
           i
           (+ (* 0.041666666666666664 (* i n)) (* n 0.16666666666666666))))))))
     (+ (* n 100.0) (* 100.0 (* i (* n (+ 0.5 (* 0.5 (/ -1.0 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.2e-54) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if ((i <= -4.1e-199) || !(i <= 1.02e-162)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else {
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.2e-54) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if ((i <= -4.1e-199) || !(i <= 1.02e-162)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else {
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.2e-54:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif (i <= -4.1e-199) or not (i <= 1.02e-162):
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))))
	else:
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.2e-54)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif ((i <= -4.1e-199) || !(i <= 1.02e-162))
		tmp = Float64(Float64(100.0 / i) * Float64(i * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666))))))));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(100.0 * Float64(i * Float64(n * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.2e-54], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[i, -4.1e-199], N[Not[LessEqual[i, 1.02e-162]], $MachinePrecision]], N[(N[(100.0 / i), $MachinePrecision] * N[(i * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(100.0 * N[(i * N[(n * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.2e-54

   1. Initial program 46.1%

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

   3. Taylor expanded in n around inf 71.2%

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

      1. expm1-define 79.2%

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

   5. Simplified 79.2%

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

   if -2.2e-54 < i < -4.10000000000000022e-199 or 1.01999999999999998e-162 < i

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf 36.7%

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

      1. expm1-define 57.8%

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

   5. Simplified 57.8%

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

      1. clear-num 57.7%

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

      2. un-div-inv 57.7%

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

      3. associate-/l/ 71.2%

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

   7. Applied egg-rr 71.2%

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

      1. /-rgt-identity 71.2%

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

      2. associate-/r/ 71.2%

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

      3. associate-/r/ 71.2%

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

      4. associate-/r/ 71.3%

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

      5. /-rgt-identity 71.3%

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

      6. *-commutative 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in i around 0 68.7%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)\right)} \]

   if -4.10000000000000022e-199 < i < 1.01999999999999998e-162

   1. Initial program 4.1%

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

   3. Taylor expanded in i around 0 95.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -4.1 \cdot 10^{-199} \lor \neg \left(i \leq 1.02 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.5 \cdot 10^{-47}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq -8 \cdot 10^{-200} \lor \neg \left(i \leq 2.6 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -7.5e-47)
   (* 100.0 (* (expm1 i) (/ n i)))
   (if (or (<= i -8e-200) (not (<= i 2.6e-165)))
     (*
      (/ 100.0 i)
      (*
       i
       (+
        n
        (*
         i
         (+
          (* n 0.5)
          (*
           i
           (+ (* 0.041666666666666664 (* i n)) (* n 0.16666666666666666))))))))
     (+ (* n 100.0) (* 100.0 (* i (* n (+ 0.5 (* 0.5 (/ -1.0 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -7.5e-47) {
		tmp = 100.0 * (expm1(i) * (n / i));
	} else if ((i <= -8e-200) || !(i <= 2.6e-165)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else {
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -7.5e-47) {
		tmp = 100.0 * (Math.expm1(i) * (n / i));
	} else if ((i <= -8e-200) || !(i <= 2.6e-165)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else {
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -7.5e-47:
		tmp = 100.0 * (math.expm1(i) * (n / i))
	elif (i <= -8e-200) or not (i <= 2.6e-165):
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))))
	else:
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -7.5e-47)
		tmp = Float64(100.0 * Float64(expm1(i) * Float64(n / i)));
	elseif ((i <= -8e-200) || !(i <= 2.6e-165))
		tmp = Float64(Float64(100.0 / i) * Float64(i * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666))))))));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(100.0 * Float64(i * Float64(n * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -7.5e-47], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[i, -8e-200], N[Not[LessEqual[i, 2.6e-165]], $MachinePrecision]], N[(N[(100.0 / i), $MachinePrecision] * N[(i * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(100.0 * N[(i * N[(n * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -7.49999999999999969e-47

   1. Initial program 46.1%

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

      1. associate-/r/ 44.6%

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

      2. add-exp-log 44.6%

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

      3. expm1-define 44.6%

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

      4. log-pow 49.4%

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

      5. log1p-define 93.6%

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in n around inf 69.9%

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

      1. expm1-define 77.8%

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

      2. associate-*l/ 77.8%

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

   7. Simplified 77.8%

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

   if -7.49999999999999969e-47 < i < -7.9999999999999999e-200 or 2.60000000000000007e-165 < i

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf 36.7%

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

      1. expm1-define 57.8%

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

   5. Simplified 57.8%

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

      1. clear-num 57.7%

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

      2. un-div-inv 57.7%

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

      3. associate-/l/ 71.2%

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

   7. Applied egg-rr 71.2%

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

      1. /-rgt-identity 71.2%

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

      2. associate-/r/ 71.2%

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

      3. associate-/r/ 71.2%

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

      4. associate-/r/ 71.3%

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

      5. /-rgt-identity 71.3%

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

      6. *-commutative 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in i around 0 68.7%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)\right)} \]

   if -7.9999999999999999e-200 < i < 2.60000000000000007e-165

   1. Initial program 4.1%

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

   3. Taylor expanded in i around 0 95.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.5 \cdot 10^{-47}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq -8 \cdot 10^{-200} \lor \neg \left(i \leq 2.6 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;n \leq -1.08 \cdot 10^{-148}:\\ \;\;\;\;100 \cdot t\_0\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (+
          n
          (*
           i
           (+
            (* n 0.5)
            (*
             i
             (+
              (* 0.041666666666666664 (* i n))
              (* n 0.16666666666666666))))))))
   (if (<= n -1.08e-148)
     (* 100.0 t_0)
     (if (<= n 3.5e-52) (/ 0.0 (/ i n)) (* (/ 100.0 i) (* i t_0))))))

C

double code(double i, double n) {
	double t_0 = n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))));
	double tmp;
	if (n <= -1.08e-148) {
		tmp = 100.0 * t_0;
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (100.0 / i) * (i * 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 * ((n * 0.5d0) + (i * ((0.041666666666666664d0 * (i * n)) + (n * 0.16666666666666666d0)))))
    if (n <= (-1.08d-148)) then
        tmp = 100.0d0 * t_0
    else if (n <= 3.5d-52) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = (100.0d0 / i) * (i * t_0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))));
	double tmp;
	if (n <= -1.08e-148) {
		tmp = 100.0 * t_0;
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (100.0 / i) * (i * t_0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))
	tmp = 0
	if n <= -1.08e-148:
		tmp = 100.0 * t_0
	elif n <= 3.5e-52:
		tmp = 0.0 / (i / n)
	else:
		tmp = (100.0 / i) * (i * t_0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666))))))
	tmp = 0.0
	if (n <= -1.08e-148)
		tmp = Float64(100.0 * t_0);
	elseif (n <= 3.5e-52)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(Float64(100.0 / i) * Float64(i * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))));
	tmp = 0.0;
	if (n <= -1.08e-148)
		tmp = 100.0 * t_0;
	elseif (n <= 3.5e-52)
		tmp = 0.0 / (i / n);
	else
		tmp = (100.0 / i) * (i * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.08e-148], N[(100.0 * t$95$0), $MachinePrecision], If[LessEqual[n, 3.5e-52], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 / i), $MachinePrecision] * N[(i * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.08000000000000006e-148

   1. Initial program 20.6%

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

   3. Taylor expanded in n around inf 36.6%

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

      1. expm1-define 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in i around 0 59.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)} \]

   if -1.08000000000000006e-148 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

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

   if 3.5e-52 < n

   1. Initial program 19.7%

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

   3. Taylor expanded in n around inf 42.1%

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

      1. expm1-define 78.2%

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

   5. Simplified 78.2%

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

      1. clear-num 78.1%

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

      2. un-div-inv 78.1%

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

      3. associate-/l/ 96.0%

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

   7. Applied egg-rr 96.0%

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

      1. /-rgt-identity 96.0%

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

      2. associate-/r/ 96.0%

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

      3. associate-/r/ 95.9%

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

      4. associate-/r/ 96.0%

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

      5. /-rgt-identity 96.0%

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

      6. *-commutative 96.0%

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

   9. Simplified 96.0%

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

   10. Taylor expanded in i around 0 85.5%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.08 \cdot 10^{-148}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-149}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8.5e-149)
   (*
    100.0
    (+
     n
     (*
      i
      (+
       (* n 0.5)
       (* i (+ (* 0.041666666666666664 (* i n)) (* n 0.16666666666666666)))))))
   (if (<= n 3.5e-52)
     (/ 0.0 (/ i n))
     (+
      (* n 100.0)
      (*
       i
       (* n (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8.5e-149) {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-8.5d-149)) then
        tmp = 100.0d0 * (n + (i * ((n * 0.5d0) + (i * ((0.041666666666666664d0 * (i * n)) + (n * 0.16666666666666666d0))))))
    else if (n <= 3.5d-52) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8.5e-149) {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8.5e-149:
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))))
	elif n <= 3.5e-52:
		tmp = 0.0 / (i / n)
	else:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8.5e-149)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666)))))));
	elseif (n <= 3.5e-52)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -8.5e-149)
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	elseif (n <= 3.5e-52)
		tmp = 0.0 / (i / n);
	else
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8.5e-149], N[(100.0 * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-52], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.5000000000000006e-149

   1. Initial program 20.6%

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

   3. Taylor expanded in n around inf 36.6%

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

      1. expm1-define 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in i around 0 59.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)} \]

   if -8.5000000000000006e-149 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

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

   if 3.5e-52 < n

   1. Initial program 19.7%

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

      1. associate-*r/ 19.7%

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

      2. sub-neg 19.7%

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

      3. distribute-rgt-in 19.7%

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

      4. metadata-eval 19.7%

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

      5. metadata-eval 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in n around inf 42.0%

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

   6. Taylor expanded in i around 0 83.6%

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

   7. Taylor expanded in n around 0 83.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-149}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-149}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.5e-149)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 3.5e-52)
     (/ 0.0 (/ i n))
     (+
      (* n 100.0)
      (*
       i
       (* n (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-149) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.5d-149)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 3.5d-52) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-149) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.5e-149:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 3.5e-52:
		tmp = 0.0 / (i / n)
	else:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.5e-149)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 3.5e-52)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.5e-149)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 3.5e-52)
		tmp = 0.0 / (i / n);
	else
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.5e-149], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-52], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.50000000000000034e-149

   1. Initial program 20.6%

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

      1. associate-*r/ 20.6%

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

      2. sub-neg 20.6%

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

      3. distribute-rgt-in 20.7%

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

      4. metadata-eval 20.7%

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

      5. metadata-eval 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in n around inf 36.6%

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

   6. Taylor expanded in i around 0 58.0%

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

   7. Taylor expanded in n around 0 59.1%

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

   if -9.50000000000000034e-149 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

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

   if 3.5e-52 < n

   1. Initial program 19.7%

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

      1. associate-*r/ 19.7%

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

      2. sub-neg 19.7%

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

      3. distribute-rgt-in 19.7%

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

      4. metadata-eval 19.7%

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

      5. metadata-eval 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in n around inf 42.0%

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

   6. Taylor expanded in i around 0 83.6%

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

   7. Taylor expanded in n around 0 83.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-149}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-149}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(i \cdot n\right) \cdot 16.666666666666668 + n \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.8e-149)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 3.5e-52)
     (/ 0.0 (/ i n))
     (+ (* n 100.0) (* i (+ (* (* i n) 16.666666666666668) (* n 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.8e-149) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (((i * n) * 16.666666666666668) + (n * 50.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.8d-149)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 3.5d-52) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = (n * 100.0d0) + (i * (((i * n) * 16.666666666666668d0) + (n * 50.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.8e-149) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (((i * n) * 16.666666666666668) + (n * 50.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.8e-149:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 3.5e-52:
		tmp = 0.0 / (i / n)
	else:
		tmp = (n * 100.0) + (i * (((i * n) * 16.666666666666668) + (n * 50.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.8e-149)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 3.5e-52)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(Float64(i * n) * 16.666666666666668) + Float64(n * 50.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.8e-149)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 3.5e-52)
		tmp = 0.0 / (i / n);
	else
		tmp = (n * 100.0) + (i * (((i * n) * 16.666666666666668) + (n * 50.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.8e-149], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-52], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(N[(i * n), $MachinePrecision] * 16.666666666666668), $MachinePrecision] + N[(n * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.8000000000000008e-149

   1. Initial program 20.6%

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

      1. associate-*r/ 20.6%

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

      2. sub-neg 20.6%

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

      3. distribute-rgt-in 20.7%

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

      4. metadata-eval 20.7%

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

      5. metadata-eval 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in n around inf 36.6%

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

   6. Taylor expanded in i around 0 58.0%

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

   7. Taylor expanded in n around 0 59.1%

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

   if -9.8000000000000008e-149 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

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

   if 3.5e-52 < n

   1. Initial program 19.7%

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

      1. associate-*r/ 19.7%

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

      2. sub-neg 19.7%

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

      3. distribute-rgt-in 19.7%

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

      4. metadata-eval 19.7%

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

      5. metadata-eval 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in n around inf 42.0%

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

   6. Taylor expanded in i around 0 80.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-149}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(i \cdot n\right) \cdot 16.666666666666668 + n \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 50 + i \cdot 16.666666666666668\\ \mathbf{if}\;n \leq -4 \cdot 10^{-148}:\\ \;\;\;\;n \cdot \left(100 + i \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 50.0 (* i 16.666666666666668))))
   (if (<= n -4e-148)
     (* n (+ 100.0 (* i t_0)))
     (if (<= n 3.5e-52) (/ 0.0 (/ i n)) (+ (* n 100.0) (* i (* n t_0)))))))

C

double code(double i, double n) {
	double t_0 = 50.0 + (i * 16.666666666666668);
	double tmp;
	if (n <= -4e-148) {
		tmp = n * (100.0 + (i * t_0));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (n * 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 = 50.0d0 + (i * 16.666666666666668d0)
    if (n <= (-4d-148)) then
        tmp = n * (100.0d0 + (i * t_0))
    else if (n <= 3.5d-52) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = (n * 100.0d0) + (i * (n * t_0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 50.0 + (i * 16.666666666666668);
	double tmp;
	if (n <= -4e-148) {
		tmp = n * (100.0 + (i * t_0));
	} else if (n <= 3.5e-52) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (n * 100.0) + (i * (n * t_0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 50.0 + (i * 16.666666666666668)
	tmp = 0
	if n <= -4e-148:
		tmp = n * (100.0 + (i * t_0))
	elif n <= 3.5e-52:
		tmp = 0.0 / (i / n)
	else:
		tmp = (n * 100.0) + (i * (n * t_0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(50.0 + Float64(i * 16.666666666666668))
	tmp = 0.0
	if (n <= -4e-148)
		tmp = Float64(n * Float64(100.0 + Float64(i * t_0)));
	elseif (n <= 3.5e-52)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 50.0 + (i * 16.666666666666668);
	tmp = 0.0;
	if (n <= -4e-148)
		tmp = n * (100.0 + (i * t_0));
	elseif (n <= 3.5e-52)
		tmp = 0.0 / (i / n);
	else
		tmp = (n * 100.0) + (i * (n * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4e-148], N[(n * N[(100.0 + N[(i * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-52], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.99999999999999974e-148

   1. Initial program 20.6%

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

      1. associate-*r/ 20.6%

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

      2. sub-neg 20.6%

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

      3. distribute-rgt-in 20.7%

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

      4. metadata-eval 20.7%

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

      5. metadata-eval 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in n around inf 36.6%

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

   6. Taylor expanded in i around 0 58.0%

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

   7. Taylor expanded in n around 0 59.1%

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

   if -3.99999999999999974e-148 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

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

   if 3.5e-52 < n

   1. Initial program 19.7%

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

      1. associate-*r/ 19.7%

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

      2. sub-neg 19.7%

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

      3. distribute-rgt-in 19.7%

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

      4. metadata-eval 19.7%

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

      5. metadata-eval 19.7%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in n around inf 42.0%

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

   6. Taylor expanded in i around 0 80.5%

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

      1. associate-*r* 80.5%

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

      2. distribute-rgt-out 80.5%

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

      3. *-commutative 80.5%

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

   8. Applied egg-rr 80.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-148}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-148} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.85e-148) (not (<= n 3.5e-52)))
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-148) || !(n <= 3.5e-52)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 0.0 / (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 <= (-1.85d-148)) .or. (.not. (n <= 3.5d-52))) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else
        tmp = 0.0d0 / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-148) || !(n <= 3.5e-52)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.85e-148) or not (n <= 3.5e-52):
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.85e-148) || !(n <= 3.5e-52))
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.85e-148) || ~((n <= 3.5e-52)))
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	else
		tmp = 0.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.85e-148], N[Not[LessEqual[n, 3.5e-52]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.85000000000000017e-148 or 3.5e-52 < n

   1. Initial program 20.2%

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

      1. associate-*r/ 20.2%

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

      2. sub-neg 20.2%

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

      3. distribute-rgt-in 20.2%

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

      4. metadata-eval 20.2%

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

      5. metadata-eval 20.2%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in n around inf 39.1%

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

   6. Taylor expanded in i around 0 68.0%

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

   7. Taylor expanded in n around 0 68.6%

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

   if -1.85000000000000017e-148 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-148} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-147} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.22e-147) (not (<= n 3.5e-52)))
   (* n (+ 100.0 (* i 50.0)))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.22e-147) || !(n <= 3.5e-52)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0 / (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 <= (-1.22d-147)) .or. (.not. (n <= 3.5d-52))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0 / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.22e-147) || !(n <= 3.5e-52)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.22e-147) or not (n <= 3.5e-52):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.22e-147) || !(n <= 3.5e-52))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.22e-147) || ~((n <= 3.5e-52)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.22e-147], N[Not[LessEqual[n, 3.5e-52]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.21999999999999995e-147 or 3.5e-52 < n

   1. Initial program 20.2%

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

      1. associate-*r/ 20.2%

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

      2. sub-neg 20.2%

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

      3. distribute-rgt-in 20.2%

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

      4. metadata-eval 20.2%

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

      5. metadata-eval 20.2%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in n around inf 39.1%

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

   6. Taylor expanded in i around 0 68.0%

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

   7. Taylor expanded in i around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. associate-*r* 65.8%

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

      3. distribute-rgt-out 65.8%

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

      4. *-commutative 65.8%

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

   9. Simplified 65.8%

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

   if -1.21999999999999995e-147 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-147} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-149} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -8.5e-149) (not (<= n 3.5e-52)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -8.5e-149) || !(n <= 3.5e-52)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8.5d-149)) .or. (.not. (n <= 3.5d-52))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -8.5e-149) || !(n <= 3.5e-52)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -8.5e-149) or not (n <= 3.5e-52):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -8.5e-149) || !(n <= 3.5e-52))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -8.5e-149) || ~((n <= 3.5e-52)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -8.5e-149], N[Not[LessEqual[n, 3.5e-52]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -8.5000000000000006e-149 or 3.5e-52 < n

   1. Initial program 20.2%

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

      1. associate-*r/ 20.2%

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

      2. sub-neg 20.2%

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

      3. distribute-rgt-in 20.2%

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

      4. metadata-eval 20.2%

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

      5. metadata-eval 20.2%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in n around inf 39.1%

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

   6. Taylor expanded in i around 0 68.0%

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

   7. Taylor expanded in i around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. associate-*r* 65.8%

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

      3. distribute-rgt-out 65.8%

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

      4. *-commutative 65.8%

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

   9. Simplified 65.8%

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

   if -8.5000000000000006e-149 < n < 3.5e-52

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in i around 0 70.5%

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

   6. Taylor expanded in i around 0 70.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-149} \lor \neg \left(n \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.6 \cdot 10^{-149} \lor \neg \left(n \leq 2.4 \cdot 10^{-83}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -8.6e-149) (not (<= n 2.4e-83))) (* n 100.0) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -8.6e-149) || !(n <= 2.4e-83)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8.6d-149)) .or. (.not. (n <= 2.4d-83))) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -8.6e-149) || !(n <= 2.4e-83)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -8.6e-149) or not (n <= 2.4e-83):
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -8.6e-149) || !(n <= 2.4e-83))
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -8.6e-149) || ~((n <= 2.4e-83)))
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -8.6e-149], N[Not[LessEqual[n, 2.4e-83]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -8.60000000000000073e-149 or 2.4000000000000001e-83 < n

   1. Initial program 20.4%

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

   3. Taylor expanded in i around 0 55.8%

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

      1. *-commutative 55.8%

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

   5. Simplified 55.8%

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

   if -8.60000000000000073e-149 < n < 2.4000000000000001e-83

   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 73.0%

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

   6. Taylor expanded in i around 0 73.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.6 \cdot 10^{-149} \lor \neg \left(n \leq 2.4 \cdot 10^{-83}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 18.7% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 28.9%

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

   1. associate-*r/ 28.9%

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

   2. sub-neg 28.9%

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

   3. distribute-rgt-in 28.9%

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

   4. metadata-eval 28.9%

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

   5. metadata-eval 28.9%

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

3. Simplified 28.9%

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

5. Taylor expanded in i around 0 21.3%

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

6. Taylor expanded in i around 0 21.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Developer target: 35.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024107 
(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))))