Compound Interest

Percentage Accurate: 29.3% → 89.7%

Time: 26.2s

Alternatives: 18

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: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) 0.0)
   (* n (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) i)))
   (* n (/ (* 100.0 (expm1 i)) i))))

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= 0.0) {
		tmp = n * (100.0 * (expm1((n * log1p((i / n)))) / i));
	} else {
		tmp = n * ((100.0 * expm1(i)) / i);
	}
	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)) <= 0.0) {
		tmp = n * (100.0 * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= 0.0:
		tmp = n * (100.0 * (math.expm1((n * math.log1p((i / n)))) / i))
	else:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	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)) <= 0.0)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	else
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	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], 0.0], 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 * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $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)) < -0.0

   1. Initial program 29.7%

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

      1. associate-*r/ 29.7%

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

      2. sub-neg 29.7%

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

      3. distribute-rgt-in 29.7%

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

      4. metadata-eval 29.7%

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

      5. metadata-eval 29.7%

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

   3. Simplified 29.7%

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

      1. metadata-eval 29.7%

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

      2. metadata-eval 29.7%

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

      3. distribute-rgt-in 29.7%

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

      4. sub-neg 29.7%

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

      5. associate-*r/ 29.7%

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

      6. associate-/r/ 29.2%

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

      7. associate-*r* 29.2%

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

      8. add-exp-log 29.2%

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

      9. expm1-define 29.2%

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

      10. log-pow 36.1%

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

      11. log1p-define 96.7%

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

   6. Applied egg-rr 96.7%

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

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

   1. Initial program 26.2%

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

      1. associate-/r/ 27.5%

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

      2. associate-*r* 27.5%

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

      3. *-commutative 27.5%

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

      4. associate-*r/ 27.5%

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

      5. sub-neg 27.5%

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

      6. distribute-lft-in 27.5%

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

      7. metadata-eval 27.5%

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

      8. metadata-eval 27.5%

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

      9. metadata-eval 27.5%

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

      10. fma-define 27.5%

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

      11. metadata-eval 27.5%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in n around inf 21.1%

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

      1. sub-neg 21.1%

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

      2. metadata-eval 21.1%

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

      3. metadata-eval 21.1%

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

      4. distribute-lft-in 21.1%

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

      5. metadata-eval 21.1%

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

      6. sub-neg 21.1%

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

      7. expm1-define 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 90.9%

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

Alternative 2: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) 0.0)
   (* 100.0 (* n (/ (expm1 (* n (log1p (/ i n)))) i)))
   (* n (/ (* 100.0 (expm1 i)) i))))

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= 0.0) {
		tmp = 100.0 * (n * (expm1((n * log1p((i / n)))) / i));
	} else {
		tmp = n * ((100.0 * expm1(i)) / i);
	}
	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)) <= 0.0) {
		tmp = 100.0 * (n * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= 0.0:
		tmp = 100.0 * (n * (math.expm1((n * math.log1p((i / n)))) / i))
	else:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	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)) <= 0.0)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	else
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	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], 0.0], 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 * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $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)) < -0.0

   1. Initial program 29.7%

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

      1. associate-/r/ 29.2%

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

      2. add-exp-log 29.2%

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

      3. expm1-define 29.2%

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

      4. log-pow 36.0%

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

      5. log1p-define 96.7%

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

   4. Applied egg-rr 96.7%

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

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

   1. Initial program 26.2%

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

      1. associate-/r/ 27.5%

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

      2. associate-*r* 27.5%

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

      3. *-commutative 27.5%

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

      4. associate-*r/ 27.5%

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

      5. sub-neg 27.5%

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

      6. distribute-lft-in 27.5%

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

      7. metadata-eval 27.5%

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

      8. metadata-eval 27.5%

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

      9. metadata-eval 27.5%

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

      10. fma-define 27.5%

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

      11. metadata-eval 27.5%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in n around inf 21.1%

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

      1. sub-neg 21.1%

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

      2. metadata-eval 21.1%

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

      3. metadata-eval 21.1%

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

      4. distribute-lft-in 21.1%

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

      5. metadata-eval 21.1%

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

      6. sub-neg 21.1%

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

      7. expm1-define 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 90.9%

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

Alternative 3: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-145}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-234}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{+37}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -2e+28)
     t_0
     (if (<= n -1.95e-145)
       (* 100.0 (* (expm1 i) (/ n i)))
       (if (<= n 2.45e-234)
         0.0
         (if (<= n 7.5e-41)
           (* 100.0 (/ i (/ i n)))
           (if (<= n 2.8e+37)
             (+
              (* n 100.0)
              (*
               i
               (*
                n
                (+
                 50.0
                 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
             t_0)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -2e+28) {
		tmp = t_0;
	} else if (n <= -1.95e-145) {
		tmp = 100.0 * (expm1(i) * (n / i));
	} else if (n <= 2.45e-234) {
		tmp = 0.0;
	} else if (n <= 7.5e-41) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.8e+37) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -2e+28) {
		tmp = t_0;
	} else if (n <= -1.95e-145) {
		tmp = 100.0 * (Math.expm1(i) * (n / i));
	} else if (n <= 2.45e-234) {
		tmp = 0.0;
	} else if (n <= 7.5e-41) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.8e+37) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -2e+28:
		tmp = t_0
	elif n <= -1.95e-145:
		tmp = 100.0 * (math.expm1(i) * (n / i))
	elif n <= 2.45e-234:
		tmp = 0.0
	elif n <= 7.5e-41:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.8e+37:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -2e+28)
		tmp = t_0;
	elseif (n <= -1.95e-145)
		tmp = Float64(100.0 * Float64(expm1(i) * Float64(n / i)));
	elseif (n <= 2.45e-234)
		tmp = 0.0;
	elseif (n <= 7.5e-41)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.8e+37)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2e+28], t$95$0, If[LessEqual[n, -1.95e-145], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.45e-234], 0.0, If[LessEqual[n, 7.5e-41], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e+37], 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], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.99999999999999992e28 or 2.7999999999999998e37 < n

   1. Initial program 21.9%

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

   3. Taylor expanded in n around inf 49.6%

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

      1. expm1-define 73.4%

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

   5. Simplified 73.4%

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

      1. clear-num 73.3%

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

      2. un-div-inv 73.4%

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

      3. associate-/l/ 95.9%

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

   7. Applied egg-rr 95.9%

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

      1. /-rgt-identity 95.9%

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

      2. associate-/r/ 95.9%

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

      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 inf 50.1%

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

       1. associate-*r/ 49.4%

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

       2. expm1-define 95.2%

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

       3. associate-*r* 95.4%

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

   12. Simplified 95.4%

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

   13. Taylor expanded in n around 0 50.1%

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

       1. expm1-define 96.1%

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

   15. Simplified 96.1%

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

   if -1.99999999999999992e28 < n < -1.95000000000000015e-145

   1. Initial program 15.4%

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

      1. associate-/r/ 15.4%

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

      2. add-exp-log 15.4%

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

      3. expm1-define 15.4%

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

      4. log-pow 38.6%

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

      5. log1p-define 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in n around inf 8.5%

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

      1. expm1-define 56.0%

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

      2. associate-*l/ 70.5%

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

   7. Simplified 70.5%

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

   if -1.95000000000000015e-145 < n < 2.45000000000000004e-234

   1. Initial program 73.2%

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

      1. associate-*r/ 73.2%

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

      2. sub-neg 73.2%

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

      3. distribute-rgt-in 73.2%

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

      4. metadata-eval 73.2%

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

      5. metadata-eval 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in i around 0 80.9%

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

   6. Taylor expanded in i around 0 80.9%

      \[\leadsto \color{blue}{0} \]

   if 2.45000000000000004e-234 < n < 7.50000000000000049e-41

   1. Initial program 23.2%

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

   3. Taylor expanded in i around 0 66.3%

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

   if 7.50000000000000049e-41 < n < 2.7999999999999998e37

   1. Initial program 17.6%

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

      1. associate-*r/ 17.6%

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

      2. sub-neg 17.6%

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

      3. distribute-rgt-in 17.6%

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

      4. metadata-eval 17.6%

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

      5. metadata-eval 17.6%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in n around inf 15.6%

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

   6. Taylor expanded in i around 0 95.8%

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

   7. Taylor expanded in n around 0 95.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-145}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-234}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{+37}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ t_1 := \frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{if}\;i \leq -5.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq -2.35 \cdot 10^{-177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{-227}:\\ \;\;\;\;n \cdot 100 + i \cdot -50\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+158}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+204}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{+273}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n))))
        (t_1 (/ (* i (* n (+ 100.0 (* i 50.0)))) i)))
   (if (<= i -5.2e-8)
     t_0
     (if (<= i -2.35e-177)
       t_1
       (if (<= i 2.9e-227)
         (+ (* n 100.0) (* i -50.0))
         (if (<= i 1.9e-53)
           t_1
           (if (<= i 1.22e+117)
             t_0
             (if (<= i 1.06e+158)
               0.0
               (if (<= i 2.2e+204)
                 (/ (* i (+ (* n 100.0) (* 50.0 (* i n)))) i)
                 (if (<= i 1.5e+273) 0.0 (* (* i n) (/ 100.0 i))))))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double t_1 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (i <= -5.2e-8) {
		tmp = t_0;
	} else if (i <= -2.35e-177) {
		tmp = t_1;
	} else if (i <= 2.9e-227) {
		tmp = (n * 100.0) + (i * -50.0);
	} else if (i <= 1.9e-53) {
		tmp = t_1;
	} else if (i <= 1.22e+117) {
		tmp = t_0;
	} else if (i <= 1.06e+158) {
		tmp = 0.0;
	} else if (i <= 2.2e+204) {
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	} else if (i <= 1.5e+273) {
		tmp = 0.0;
	} else {
		tmp = (i * n) * (100.0 / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double t_1 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (i <= -5.2e-8) {
		tmp = t_0;
	} else if (i <= -2.35e-177) {
		tmp = t_1;
	} else if (i <= 2.9e-227) {
		tmp = (n * 100.0) + (i * -50.0);
	} else if (i <= 1.9e-53) {
		tmp = t_1;
	} else if (i <= 1.22e+117) {
		tmp = t_0;
	} else if (i <= 1.06e+158) {
		tmp = 0.0;
	} else if (i <= 2.2e+204) {
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	} else if (i <= 1.5e+273) {
		tmp = 0.0;
	} else {
		tmp = (i * n) * (100.0 / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	t_1 = (i * (n * (100.0 + (i * 50.0)))) / i
	tmp = 0
	if i <= -5.2e-8:
		tmp = t_0
	elif i <= -2.35e-177:
		tmp = t_1
	elif i <= 2.9e-227:
		tmp = (n * 100.0) + (i * -50.0)
	elif i <= 1.9e-53:
		tmp = t_1
	elif i <= 1.22e+117:
		tmp = t_0
	elif i <= 1.06e+158:
		tmp = 0.0
	elif i <= 2.2e+204:
		tmp = (i * ((n * 100.0) + (50.0 * (i * n)))) / i
	elif i <= 1.5e+273:
		tmp = 0.0
	else:
		tmp = (i * n) * (100.0 / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	t_1 = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i)
	tmp = 0.0
	if (i <= -5.2e-8)
		tmp = t_0;
	elseif (i <= -2.35e-177)
		tmp = t_1;
	elseif (i <= 2.9e-227)
		tmp = Float64(Float64(n * 100.0) + Float64(i * -50.0));
	elseif (i <= 1.9e-53)
		tmp = t_1;
	elseif (i <= 1.22e+117)
		tmp = t_0;
	elseif (i <= 1.06e+158)
		tmp = 0.0;
	elseif (i <= 2.2e+204)
		tmp = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)))) / i);
	elseif (i <= 1.5e+273)
		tmp = 0.0;
	else
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[i, -5.2e-8], t$95$0, If[LessEqual[i, -2.35e-177], t$95$1, If[LessEqual[i, 2.9e-227], N[(N[(n * 100.0), $MachinePrecision] + N[(i * -50.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.9e-53], t$95$1, If[LessEqual[i, 1.22e+117], t$95$0, If[LessEqual[i, 1.06e+158], 0.0, If[LessEqual[i, 2.2e+204], N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 1.5e+273], 0.0, N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if i < -5.2000000000000002e-8 or 1.8999999999999999e-53 < i < 1.22000000000000004e117

   1. Initial program 43.4%

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

   3. Taylor expanded in n around inf 71.3%

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

      1. expm1-define 76.6%

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

   5. Simplified 76.6%

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

   if -5.2000000000000002e-8 < i < -2.34999999999999983e-177 or 2.90000000000000011e-227 < i < 1.8999999999999999e-53

   1. Initial program 15.3%

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

   3. Taylor expanded in n around inf 16.2%

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

      1. expm1-define 64.6%

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

   5. Simplified 64.6%

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

      1. clear-num 64.5%

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

      2. un-div-inv 64.4%

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

      3. associate-/l/ 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. /-rgt-identity 88.9%

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

      2. associate-/r/ 88.9%

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

      3. associate-/r/ 88.8%

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

      4. associate-/r/ 89.0%

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

      5. /-rgt-identity 89.0%

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

      6. *-commutative 89.0%

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

   9. Simplified 89.0%

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

   10. Taylor expanded in i around inf 16.4%

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

       1. associate-*r/ 16.4%

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

       2. expm1-define 88.9%

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

       3. associate-*r* 89.2%

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

   12. Simplified 89.2%

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

   13. Taylor expanded in i around 0 89.2%

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

       1. associate-*r* 89.2%

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

       2. distribute-rgt-out 89.2%

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

       3. *-commutative 89.2%

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

   15. Applied egg-rr 89.2%

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

   if -2.34999999999999983e-177 < i < 2.90000000000000011e-227

   1. Initial program 4.5%

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

      1. associate-*r/ 4.5%

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

      2. sub-neg 4.5%

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

      3. distribute-rgt-in 4.5%

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

      4. metadata-eval 4.5%

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

      5. metadata-eval 4.5%

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

   3. Simplified 4.5%

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

      1. metadata-eval 4.5%

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

      2. metadata-eval 4.5%

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

      3. distribute-rgt-in 4.5%

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

      4. sub-neg 4.5%

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

      5. associate-*r/ 4.5%

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

      6. associate-/r/ 5.3%

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

      7. associate-*r* 5.3%

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

      8. add-exp-log 5.3%

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

      9. expm1-define 5.3%

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

      10. log-pow 11.6%

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

      11. log1p-define 59.9%

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

   6. Applied egg-rr 59.9%

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

   7. Taylor expanded in i around 0 93.4%

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

      1. sub-neg 93.4%

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

      2. associate-*r/ 93.4%

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

      3. metadata-eval 93.4%

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

      4. sub-neg 93.4%

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

   9. Simplified 93.4%

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

   10. Taylor expanded in n around 0 93.4%

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

       1. associate-*r/ 93.4%

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

       2. *-commutative 93.4%

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

       3. associate-/l* 93.4%

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

   12. Simplified 93.4%

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

   13. Taylor expanded in i around 0 93.4%

       \[\leadsto \color{blue}{-50 \cdot i + 100 \cdot n} \]

   if 1.22000000000000004e117 < i < 1.06e158 or 2.20000000000000011e204 < i < 1.5e273

   1. Initial program 38.5%

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

      1. associate-*r/ 38.5%

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

      2. sub-neg 38.5%

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

      3. distribute-rgt-in 38.5%

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

      4. metadata-eval 38.5%

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

      5. metadata-eval 38.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in i around 0 67.5%

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

   6. Taylor expanded in i around 0 67.5%

      \[\leadsto \color{blue}{0} \]

   if 1.06e158 < i < 2.20000000000000011e204

   1. Initial program 62.9%

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

   3. Taylor expanded in n around inf 75.3%

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

      1. expm1-define 75.3%

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

   5. Simplified 75.3%

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

      1. clear-num 75.3%

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

      2. un-div-inv 75.3%

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

      3. associate-/l/ 75.3%

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

   7. Applied egg-rr 75.3%

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

      1. /-rgt-identity 75.3%

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

      2. associate-/r/ 75.3%

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

      3. associate-/r/ 75.3%

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

      4. associate-/r/ 75.3%

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

      5. /-rgt-identity 75.3%

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

      6. *-commutative 75.3%

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

   9. Simplified 75.3%

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

   10. Taylor expanded in i around inf 75.3%

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

       1. associate-*r/ 75.3%

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

       2. expm1-define 75.3%

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

       3. associate-*r* 75.3%

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

   12. Simplified 75.3%

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

   13. Taylor expanded in i around 0 75.4%

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

   if 1.5e273 < i

   1. Initial program 75.0%

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

   3. Taylor expanded in n around inf 50.2%

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

      1. expm1-define 50.2%

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

   5. Simplified 50.2%

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

      1. clear-num 50.2%

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

      2. un-div-inv 50.2%

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

      3. associate-/l/ 50.6%

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

   7. Applied egg-rr 50.6%

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

      1. /-rgt-identity 50.6%

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

      2. associate-/r/ 50.6%

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

      3. associate-/r/ 50.6%

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

      4. associate-/r/ 50.6%

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

      5. /-rgt-identity 50.6%

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

      6. *-commutative 50.6%

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

   9. Simplified 50.6%

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

   10. Taylor expanded in i around 0 51.5%

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

       1. *-commutative 51.5%

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

   12. Simplified 51.5%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(n \cdot i\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.2 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -2.35 \cdot 10^{-177}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{-227}:\\ \;\;\;\;n \cdot 100 + i \cdot -50\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{+117}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+158}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+204}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{+273}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ t_1 := \frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \mathbf{if}\;i \leq -0.235:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq -2.15 \cdot 10^{-178}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + \left(i \cdot n\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{-237}:\\ \;\;\;\;n \cdot 100 + i \cdot -50\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+158}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.25 \cdot 10^{+204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+273}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* (expm1 i) (/ n i))))
        (t_1 (/ (* i (+ (* n 100.0) (* 50.0 (* i n)))) i)))
   (if (<= i -0.235)
     t_0
     (if (<= i -2.15e-178)
       (*
        (/ 100.0 i)
        (*
         i
         (+
          n
          (*
           i
           (+
            (* n 0.5)
            (* (* i n) (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
       (if (<= i 8.5e-237)
         (+ (* n 100.0) (* i -50.0))
         (if (<= i 3e-36)
           t_1
           (if (<= i 1.22e+117)
             t_0
             (if (<= i 1.06e+158)
               0.0
               (if (<= i 2.25e+204)
                 t_1
                 (if (<= i 2.8e+273) 0.0 (* (* i n) (/ 100.0 i))))))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) * (n / i));
	double t_1 = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	double tmp;
	if (i <= -0.235) {
		tmp = t_0;
	} else if (i <= -2.15e-178) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + ((i * n) * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	} else if (i <= 8.5e-237) {
		tmp = (n * 100.0) + (i * -50.0);
	} else if (i <= 3e-36) {
		tmp = t_1;
	} else if (i <= 1.22e+117) {
		tmp = t_0;
	} else if (i <= 1.06e+158) {
		tmp = 0.0;
	} else if (i <= 2.25e+204) {
		tmp = t_1;
	} else if (i <= 2.8e+273) {
		tmp = 0.0;
	} else {
		tmp = (i * n) * (100.0 / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) * (n / i));
	double t_1 = (i * ((n * 100.0) + (50.0 * (i * n)))) / i;
	double tmp;
	if (i <= -0.235) {
		tmp = t_0;
	} else if (i <= -2.15e-178) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + ((i * n) * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	} else if (i <= 8.5e-237) {
		tmp = (n * 100.0) + (i * -50.0);
	} else if (i <= 3e-36) {
		tmp = t_1;
	} else if (i <= 1.22e+117) {
		tmp = t_0;
	} else if (i <= 1.06e+158) {
		tmp = 0.0;
	} else if (i <= 2.25e+204) {
		tmp = t_1;
	} else if (i <= 2.8e+273) {
		tmp = 0.0;
	} else {
		tmp = (i * n) * (100.0 / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) * (n / i))
	t_1 = (i * ((n * 100.0) + (50.0 * (i * n)))) / i
	tmp = 0
	if i <= -0.235:
		tmp = t_0
	elif i <= -2.15e-178:
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + ((i * n) * (0.16666666666666666 + (i * 0.041666666666666664)))))))
	elif i <= 8.5e-237:
		tmp = (n * 100.0) + (i * -50.0)
	elif i <= 3e-36:
		tmp = t_1
	elif i <= 1.22e+117:
		tmp = t_0
	elif i <= 1.06e+158:
		tmp = 0.0
	elif i <= 2.25e+204:
		tmp = t_1
	elif i <= 2.8e+273:
		tmp = 0.0
	else:
		tmp = (i * n) * (100.0 / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) * Float64(n / i)))
	t_1 = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)))) / i)
	tmp = 0.0
	if (i <= -0.235)
		tmp = t_0;
	elseif (i <= -2.15e-178)
		tmp = Float64(Float64(100.0 / i) * Float64(i * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(Float64(i * n) * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))));
	elseif (i <= 8.5e-237)
		tmp = Float64(Float64(n * 100.0) + Float64(i * -50.0));
	elseif (i <= 3e-36)
		tmp = t_1;
	elseif (i <= 1.22e+117)
		tmp = t_0;
	elseif (i <= 1.06e+158)
		tmp = 0.0;
	elseif (i <= 2.25e+204)
		tmp = t_1;
	elseif (i <= 2.8e+273)
		tmp = 0.0;
	else
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[i, -0.235], t$95$0, If[LessEqual[i, -2.15e-178], N[(N[(100.0 / i), $MachinePrecision] * N[(i * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.5e-237], N[(N[(n * 100.0), $MachinePrecision] + N[(i * -50.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3e-36], t$95$1, If[LessEqual[i, 1.22e+117], t$95$0, If[LessEqual[i, 1.06e+158], 0.0, If[LessEqual[i, 2.25e+204], t$95$1, If[LessEqual[i, 2.8e+273], 0.0, N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if i < -0.23499999999999999 or 3.0000000000000002e-36 < i < 1.22000000000000004e117

   1. Initial program 45.3%

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

      1. associate-/r/ 44.1%

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

      2. add-exp-log 44.1%

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

      3. expm1-define 44.1%

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

      4. log-pow 46.8%

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

      5. log1p-define 94.3%

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in n around inf 74.5%

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

      1. expm1-define 77.5%

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

      2. associate-*l/ 76.5%

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

   7. Simplified 76.5%

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

   if -0.23499999999999999 < i < -2.15e-178

   1. Initial program 15.5%

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

   3. Taylor expanded in n around inf 19.5%

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

      1. expm1-define 63.7%

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

   5. Simplified 63.7%

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

      1. clear-num 63.5%

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

      2. un-div-inv 63.5%

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

      3. associate-/l/ 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. /-rgt-identity 87.5%

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

      2. associate-/r/ 87.5%

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

      3. associate-/r/ 87.4%

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

      4. associate-/r/ 87.7%

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

      5. /-rgt-identity 87.7%

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

      6. *-commutative 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in i around 0 87.0%

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

   11. Taylor expanded in n around 0 87.0%

       \[\leadsto \frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + \color{blue}{i \cdot \left(n \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot i\right)\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* 87.0%

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

       2. *-commutative 87.0%

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

       3. *-commutative 87.0%

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

   13. Simplified 87.0%

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

   if -2.15e-178 < i < 8.49999999999999951e-237

   1. Initial program 4.5%

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

      1. associate-*r/ 4.5%

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

      2. sub-neg 4.5%

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

      3. distribute-rgt-in 4.5%

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

      4. metadata-eval 4.5%

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

      5. metadata-eval 4.5%

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

   3. Simplified 4.5%

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

      1. metadata-eval 4.5%

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

      2. metadata-eval 4.5%

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

      3. distribute-rgt-in 4.5%

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

      4. sub-neg 4.5%

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

      5. associate-*r/ 4.5%

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

      6. associate-/r/ 5.3%

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

      7. associate-*r* 5.3%

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

      8. add-exp-log 5.3%

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

      9. expm1-define 5.3%

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

      10. log-pow 10.2%

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

      11. log1p-define 58.3%

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

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in i around 0 94.5%

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

      1. sub-neg 94.5%

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

      2. associate-*r/ 94.5%

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

      3. metadata-eval 94.5%

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

      4. sub-neg 94.5%

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

   9. Simplified 94.5%

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

   10. Taylor expanded in n around 0 94.5%

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

       1. associate-*r/ 94.5%

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

       2. *-commutative 94.5%

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

       3. associate-/l* 94.5%

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

   12. Simplified 94.5%

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

   13. Taylor expanded in i around 0 94.5%

       \[\leadsto \color{blue}{-50 \cdot i + 100 \cdot n} \]

   if 8.49999999999999951e-237 < i < 3.0000000000000002e-36 or 1.06e158 < i < 2.25000000000000001e204

   1. Initial program 24.8%

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

   3. Taylor expanded in n around inf 26.7%

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

      1. expm1-define 66.9%

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

   5. Simplified 66.9%

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

      1. clear-num 66.8%

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

      2. un-div-inv 66.7%

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

      3. associate-/l/ 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. /-rgt-identity 83.2%

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

      2. associate-/r/ 83.2%

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

      3. associate-/r/ 83.2%

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

      4. associate-/r/ 83.2%

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

      5. /-rgt-identity 83.2%

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

      6. *-commutative 83.2%

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

   9. Simplified 83.2%

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

   10. Taylor expanded in i around inf 26.9%

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

       1. associate-*r/ 26.9%

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

       2. expm1-define 83.4%

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

       3. associate-*r* 83.5%

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

   12. Simplified 83.5%

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

   13. Taylor expanded in i around 0 83.5%

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

   if 1.22000000000000004e117 < i < 1.06e158 or 2.25000000000000001e204 < i < 2.80000000000000018e273

   1. Initial program 38.5%

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

      1. associate-*r/ 38.5%

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

      2. sub-neg 38.5%

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

      3. distribute-rgt-in 38.5%

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

      4. metadata-eval 38.5%

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

      5. metadata-eval 38.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in i around 0 67.5%

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

   6. Taylor expanded in i around 0 67.5%

      \[\leadsto \color{blue}{0} \]

   if 2.80000000000000018e273 < i

   1. Initial program 75.0%

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

   3. Taylor expanded in n around inf 50.2%

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

      1. expm1-define 50.2%

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

   5. Simplified 50.2%

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

      1. clear-num 50.2%

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

      2. un-div-inv 50.2%

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

      3. associate-/l/ 50.6%

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

   7. Applied egg-rr 50.6%

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

      1. /-rgt-identity 50.6%

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

      2. associate-/r/ 50.6%

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

      3. associate-/r/ 50.6%

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

      4. associate-/r/ 50.6%

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

      5. /-rgt-identity 50.6%

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

      6. *-commutative 50.6%

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

   9. Simplified 50.6%

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

   10. Taylor expanded in i around 0 51.5%

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

       1. *-commutative 51.5%

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

   12. Simplified 51.5%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(n \cdot i\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.235:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq -2.15 \cdot 10^{-178}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + \left(i \cdot n\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{-237}:\\ \;\;\;\;n \cdot 100 + i \cdot -50\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-36}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{+117}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 1.06 \cdot 10^{+158}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.25 \cdot 10^{+204}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + 50 \cdot \left(i \cdot n\right)\right)}{i}\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+273}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{-238}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-216}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 i)) i))))
   (if (<= n -1.9e-238)
     t_0
     (if (<= n 2.7e-216)
       (/ 0.0 (/ i n))
       (if (<= n 7.5e-41) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1(i)) / i);
	double tmp;
	if (n <= -1.9e-238) {
		tmp = t_0;
	} else if (n <= 2.7e-216) {
		tmp = 0.0 / (i / n);
	} else if (n <= 7.5e-41) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 * Math.expm1(i)) / i);
	double tmp;
	if (n <= -1.9e-238) {
		tmp = t_0;
	} else if (n <= 2.7e-216) {
		tmp = 0.0 / (i / n);
	} else if (n <= 7.5e-41) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 * math.expm1(i)) / i)
	tmp = 0
	if n <= -1.9e-238:
		tmp = t_0
	elif n <= 2.7e-216:
		tmp = 0.0 / (i / n)
	elif n <= 7.5e-41:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
	tmp = 0.0
	if (n <= -1.9e-238)
		tmp = t_0;
	elseif (n <= 2.7e-216)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 7.5e-41)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.9e-238], t$95$0, If[LessEqual[n, 2.7e-216], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e-41], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.8999999999999998e-238 or 7.50000000000000049e-41 < n

   1. Initial program 22.1%

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

      1. associate-/r/ 22.3%

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

      2. associate-*r* 22.3%

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

      3. *-commutative 22.3%

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

      4. associate-*r/ 22.3%

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

      5. sub-neg 22.3%

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

      6. distribute-lft-in 22.3%

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

      7. metadata-eval 22.3%

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

      8. metadata-eval 22.3%

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

      9. metadata-eval 22.3%

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

      10. fma-define 22.3%

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

      11. metadata-eval 22.3%

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

   3. Simplified 22.3%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
   4. 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 39.5%

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

      5. metadata-eval 39.5%

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

      6. sub-neg 39.5%

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

      7. expm1-define 89.5%

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

   7. Simplified 89.5%

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

   if -1.8999999999999998e-238 < n < 2.6999999999999999e-216

   1. Initial program 70.8%

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

      1. associate-*r/ 70.8%

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

      2. sub-neg 70.8%

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

      3. distribute-rgt-in 70.8%

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

      4. metadata-eval 70.8%

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

      5. metadata-eval 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in i around 0 86.0%

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

   if 2.6999999999999999e-216 < n < 7.50000000000000049e-41

   1. Initial program 26.6%

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

   3. Taylor expanded in i around 0 63.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-238}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-216}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + \left(i \cdot n\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-118}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (/ 100.0 i)
          (*
           i
           (+
            n
            (*
             i
             (+
              (* n 0.5)
              (*
               (* i n)
               (+ 0.16666666666666666 (* i 0.041666666666666664))))))))))
   (if (<= n -6.2e+77)
     t_0
     (if (<= n -4.8e-218)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 6e-118) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = (100.0 / i) * (i * (n + (i * ((n * 0.5) + ((i * n) * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	double tmp;
	if (n <= -6.2e+77) {
		tmp = t_0;
	} else if (n <= -4.8e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 6e-118) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (100.0d0 / i) * (i * (n + (i * ((n * 0.5d0) + ((i * n) * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))
    if (n <= (-6.2d+77)) then
        tmp = t_0
    else if (n <= (-4.8d-218)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 6d-118) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 / i) * (i * (n + (i * ((n * 0.5) + ((i * n) * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	double tmp;
	if (n <= -6.2e+77) {
		tmp = t_0;
	} else if (n <= -4.8e-218) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 6e-118) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 / i) * (i * (n + (i * ((n * 0.5) + ((i * n) * (0.16666666666666666 + (i * 0.041666666666666664)))))))
	tmp = 0
	if n <= -6.2e+77:
		tmp = t_0
	elif n <= -4.8e-218:
		tmp = 100.0 * (i / (i / n))
	elif n <= 6e-118:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 / i) * Float64(i * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(Float64(i * n) * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))))
	tmp = 0.0
	if (n <= -6.2e+77)
		tmp = t_0;
	elseif (n <= -4.8e-218)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 6e-118)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 / i) * (i * (n + (i * ((n * 0.5) + ((i * n) * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	tmp = 0.0;
	if (n <= -6.2e+77)
		tmp = t_0;
	elseif (n <= -4.8e-218)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 6e-118)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 / i), $MachinePrecision] * N[(i * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e+77], t$95$0, If[LessEqual[n, -4.8e-218], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6e-118], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.19999999999999997e77 or 6.00000000000000035e-118 < n

   1. Initial program 19.0%

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

   3. Taylor expanded in n around inf 39.8%

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

      1. expm1-define 71.1%

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

   5. Simplified 71.1%

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

      1. clear-num 71.0%

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

      2. un-div-inv 71.0%

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

      3. associate-/l/ 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. /-rgt-identity 90.2%

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

      2. associate-/r/ 90.2%

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

      3. associate-/r/ 90.1%

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

      4. associate-/r/ 90.3%

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

      5. /-rgt-identity 90.3%

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

      6. *-commutative 90.3%

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

   9. Simplified 90.3%

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

   10. Taylor expanded in i around 0 73.1%

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

   11. Taylor expanded in n around 0 73.1%

       \[\leadsto \frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + \color{blue}{i \cdot \left(n \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot i\right)\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* 73.1%

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

       2. *-commutative 73.1%

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

       3. *-commutative 73.1%

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

   13. Simplified 73.1%

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

   if -6.19999999999999997e77 < n < -4.8000000000000002e-218

   1. Initial program 32.2%

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

   3. Taylor expanded in i around 0 55.8%

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

   if -4.8000000000000002e-218 < n < 6.00000000000000035e-118

   1. Initial program 55.6%

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

      1. associate-*r/ 55.6%

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

      2. sub-neg 55.6%

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

      3. distribute-rgt-in 55.6%

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

      4. metadata-eval 55.6%

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

      5. metadata-eval 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in i around 0 77.6%

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

   6. Taylor expanded in i around 0 77.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + \left(i \cdot n\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-118}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + \left(i \cdot n\right) \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-148}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(\left(i \cdot n\right) \cdot 0.041666666666666664 + n \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -8.2 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -4.7 \cdot 10^{-225}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.9e-148)
   (*
    100.0
    (+
     n
     (*
      i
      (+
       (* n 0.5)
       (* i (+ (* (* i n) 0.041666666666666664) (* n 0.16666666666666666)))))))
   (if (<= n -8.2e-209)
     0.0
     (if (<= n -4.7e-225)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 2.4e-83)
         (/ 0.0 (/ i n))
         (*
          100.0
          (*
           n
           (+
            1.0
            (*
             i
             (+
              0.5
              (*
               i
               (+ 0.16666666666666666 (* i 0.041666666666666664)))))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.9e-148) {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * (((i * n) * 0.041666666666666664) + (n * 0.16666666666666666))))));
	} else if (n <= -8.2e-209) {
		tmp = 0.0;
	} else if (n <= -4.7e-225) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.4e-83) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.9d-148)) then
        tmp = 100.0d0 * (n + (i * ((n * 0.5d0) + (i * (((i * n) * 0.041666666666666664d0) + (n * 0.16666666666666666d0))))))
    else if (n <= (-8.2d-209)) then
        tmp = 0.0d0
    else if (n <= (-4.7d-225)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 2.4d-83) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.9e-148) {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * (((i * n) * 0.041666666666666664) + (n * 0.16666666666666666))))));
	} else if (n <= -8.2e-209) {
		tmp = 0.0;
	} else if (n <= -4.7e-225) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.4e-83) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.9e-148:
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * (((i * n) * 0.041666666666666664) + (n * 0.16666666666666666))))))
	elif n <= -8.2e-209:
		tmp = 0.0
	elif n <= -4.7e-225:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.4e-83:
		tmp = 0.0 / (i / n)
	else:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.9e-148)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(Float64(i * n) * 0.041666666666666664) + Float64(n * 0.16666666666666666)))))));
	elseif (n <= -8.2e-209)
		tmp = 0.0;
	elseif (n <= -4.7e-225)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.4e-83)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.9e-148)
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * (((i * n) * 0.041666666666666664) + (n * 0.16666666666666666))))));
	elseif (n <= -8.2e-209)
		tmp = 0.0;
	elseif (n <= -4.7e-225)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 2.4e-83)
		tmp = 0.0 / (i / n);
	else
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.9e-148], N[(100.0 * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(N[(i * n), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -8.2e-209], 0.0, If[LessEqual[n, -4.7e-225], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e-83], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.90000000000000007e-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 -1.90000000000000007e-148 < n < -8.19999999999999955e-209

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. sub-neg 73.1%

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

      3. distribute-rgt-in 73.1%

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

      4. metadata-eval 73.1%

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

      5. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in i around 0 73.1%

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

   6. Taylor expanded in i around 0 73.1%

      \[\leadsto \color{blue}{0} \]

   if -8.19999999999999955e-209 < n < -4.70000000000000014e-225

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 69.0%

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

   if -4.70000000000000014e-225 < 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 76.6%

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

   if 2.4000000000000001e-83 < n

   1. Initial program 20.1%

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

   3. Taylor expanded in n around inf 40.3%

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

      1. expm1-define 75.9%

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

   5. Simplified 75.9%

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

      1. clear-num 75.8%

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

      2. un-div-inv 75.8%

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

      3. associate-/l/ 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. /-rgt-identity 92.9%

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

      2. associate-/r/ 92.9%

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

      3. associate-/r/ 92.8%

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

      4. associate-/r/ 92.9%

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

      5. /-rgt-identity 92.9%

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

      6. *-commutative 92.9%

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

   9. Simplified 92.9%

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

   10. Taylor expanded in i around 0 82.9%

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

   11. Taylor expanded in n around 0 81.0%

       \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot i\right)\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-148}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(\left(i \cdot n\right) \cdot 0.041666666666666664 + n \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -8.2 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -4.7 \cdot 10^{-225}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{if}\;n \leq -1.22 \cdot 10^{-148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -7 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          100.0
          (*
           n
           (+
            1.0
            (*
             i
             (+
              0.5
              (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))))
   (if (<= n -1.22e-148)
     t_0
     (if (<= n -4.8e-209)
       0.0
       (if (<= n -7e-224)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 2.4e-83) (/ 0.0 (/ i n)) t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	double tmp;
	if (n <= -1.22e-148) {
		tmp = t_0;
	} else if (n <= -4.8e-209) {
		tmp = 0.0;
	} else if (n <= -7e-224) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.4e-83) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))
    if (n <= (-1.22d-148)) then
        tmp = t_0
    else if (n <= (-4.8d-209)) then
        tmp = 0.0d0
    else if (n <= (-7d-224)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 2.4d-83) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	double tmp;
	if (n <= -1.22e-148) {
		tmp = t_0;
	} else if (n <= -4.8e-209) {
		tmp = 0.0;
	} else if (n <= -7e-224) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.4e-83) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))
	tmp = 0
	if n <= -1.22e-148:
		tmp = t_0
	elif n <= -4.8e-209:
		tmp = 0.0
	elif n <= -7e-224:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.4e-83:
		tmp = 0.0 / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))))
	tmp = 0.0
	if (n <= -1.22e-148)
		tmp = t_0;
	elseif (n <= -4.8e-209)
		tmp = 0.0;
	elseif (n <= -7e-224)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.4e-83)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))));
	tmp = 0.0;
	if (n <= -1.22e-148)
		tmp = t_0;
	elseif (n <= -4.8e-209)
		tmp = 0.0;
	elseif (n <= -7e-224)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 2.4e-83)
		tmp = 0.0 / (i / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.22e-148], t$95$0, If[LessEqual[n, -4.8e-209], 0.0, If[LessEqual[n, -7e-224], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e-83], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.21999999999999992e-148 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 n around inf 38.3%

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

      1. expm1-define 73.7%

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

   5. Simplified 73.7%

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

      1. clear-num 73.6%

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

      2. un-div-inv 73.6%

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

      3. associate-/l/ 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. /-rgt-identity 87.5%

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

      2. associate-/r/ 87.5%

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

      3. associate-/r/ 87.0%

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

      4. associate-/r/ 87.2%

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

      5. /-rgt-identity 87.2%

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

      6. *-commutative 87.2%

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

   9. Simplified 87.2%

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

   10. Taylor expanded in i around 0 67.8%

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

   11. Taylor expanded in n around 0 69.1%

       \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot i\right)\right)\right)\right)} \]

   if -1.21999999999999992e-148 < n < -4.8000000000000002e-209

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. sub-neg 73.1%

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

      3. distribute-rgt-in 73.1%

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

      4. metadata-eval 73.1%

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

      5. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in i around 0 73.1%

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

   6. Taylor expanded in i around 0 73.1%

      \[\leadsto \color{blue}{0} \]

   if -4.8000000000000002e-209 < n < -7.00000000000000037e-224

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 69.0%

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

   if -7.00000000000000037e-224 < 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 76.6%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-148}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -7 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{-148}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-221}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-252}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= n -2.4e-148)
     (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
     (if (<= n -9.5e-209)
       0.0
       (if (<= n -3.6e-221)
         t_0
         (if (<= n 4e-252)
           (/ 0.0 (/ i n))
           (if (<= n 1.45e-92)
             t_0
             (/ (* i (* n (+ 100.0 (* i 50.0)))) i))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -2.4e-148) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= -9.5e-209) {
		tmp = 0.0;
	} else if (n <= -3.6e-221) {
		tmp = t_0;
	} else if (n <= 4e-252) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.45e-92) {
		tmp = t_0;
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (i / (i / n))
    if (n <= (-2.4d-148)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= (-9.5d-209)) then
        tmp = 0.0d0
    else if (n <= (-3.6d-221)) then
        tmp = t_0
    else if (n <= 4d-252) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 1.45d-92) then
        tmp = t_0
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -2.4e-148) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= -9.5e-209) {
		tmp = 0.0;
	} else if (n <= -3.6e-221) {
		tmp = t_0;
	} else if (n <= 4e-252) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.45e-92) {
		tmp = t_0;
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -2.4e-148:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= -9.5e-209:
		tmp = 0.0
	elif n <= -3.6e-221:
		tmp = t_0
	elif n <= 4e-252:
		tmp = 0.0 / (i / n)
	elif n <= 1.45e-92:
		tmp = t_0
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -2.4e-148)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= -9.5e-209)
		tmp = 0.0;
	elseif (n <= -3.6e-221)
		tmp = t_0;
	elseif (n <= 4e-252)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.45e-92)
		tmp = t_0;
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -2.4e-148)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= -9.5e-209)
		tmp = 0.0;
	elseif (n <= -3.6e-221)
		tmp = t_0;
	elseif (n <= 4e-252)
		tmp = 0.0 / (i / n);
	elseif (n <= 1.45e-92)
		tmp = t_0;
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.4e-148], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -9.5e-209], 0.0, If[LessEqual[n, -3.6e-221], t$95$0, If[LessEqual[n, 4e-252], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.45e-92], t$95$0, N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -2.4000000000000001e-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 -2.4000000000000001e-148 < n < -9.50000000000000028e-209

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. sub-neg 73.1%

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

      3. distribute-rgt-in 73.1%

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

      4. metadata-eval 73.1%

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

      5. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in i around 0 73.1%

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

   6. Taylor expanded in i around 0 73.1%

      \[\leadsto \color{blue}{0} \]

   if -9.50000000000000028e-209 < n < -3.60000000000000011e-221 or 3.99999999999999977e-252 < n < 1.44999999999999992e-92

   1. Initial program 28.6%

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

   3. Taylor expanded in i around 0 63.3%

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

   if -3.60000000000000011e-221 < n < 3.99999999999999977e-252

   1. Initial program 80.8%

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

      1. associate-*r/ 80.8%

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

      2. sub-neg 80.8%

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

      3. distribute-rgt-in 80.8%

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

      4. metadata-eval 80.8%

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

      5. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in i around 0 92.8%

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

   if 1.44999999999999992e-92 < n

   1. Initial program 20.5%

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

   3. Taylor expanded in n around inf 39.0%

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

      1. expm1-define 73.4%

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

   5. Simplified 73.4%

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

      1. clear-num 73.3%

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

      2. un-div-inv 73.3%

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

      3. associate-/l/ 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. /-rgt-identity 89.8%

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

      2. associate-/r/ 89.9%

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

      3. associate-/r/ 89.7%

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

      4. associate-/r/ 89.9%

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

      5. /-rgt-identity 89.9%

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

      6. *-commutative 89.9%

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

   9. Simplified 89.9%

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

   10. Taylor expanded in i around inf 39.4%

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

       1. associate-*r/ 39.3%

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

       2. expm1-define 89.8%

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

       3. associate-*r* 90.0%

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

   12. Simplified 90.0%

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

   13. Taylor expanded in i around 0 77.1%

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

       1. associate-*r* 77.1%

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

       2. distribute-rgt-out 77.1%

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

       3. *-commutative 77.1%

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

   15. Applied egg-rr 77.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{-148}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-221}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-252}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-92}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{if}\;n \leq -9.2 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.9 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -2.5 \cdot 10^{-233}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (+
           100.0
           (*
            i
            (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))))
   (if (<= n -9.2e-149)
     t_0
     (if (<= n -4.9e-209)
       0.0
       (if (<= n -2.5e-233)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 2.4e-83) (/ 0.0 (/ i n)) t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	double tmp;
	if (n <= -9.2e-149) {
		tmp = t_0;
	} else if (n <= -4.9e-209) {
		tmp = 0.0;
	} else if (n <= -2.5e-233) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.4e-83) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    if (n <= (-9.2d-149)) then
        tmp = t_0
    else if (n <= (-4.9d-209)) then
        tmp = 0.0d0
    else if (n <= (-2.5d-233)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 2.4d-83) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	double tmp;
	if (n <= -9.2e-149) {
		tmp = t_0;
	} else if (n <= -4.9e-209) {
		tmp = 0.0;
	} else if (n <= -2.5e-233) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.4e-83) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	tmp = 0
	if n <= -9.2e-149:
		tmp = t_0
	elif n <= -4.9e-209:
		tmp = 0.0
	elif n <= -2.5e-233:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.4e-83:
		tmp = 0.0 / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))
	tmp = 0.0
	if (n <= -9.2e-149)
		tmp = t_0;
	elseif (n <= -4.9e-209)
		tmp = 0.0;
	elseif (n <= -2.5e-233)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.4e-83)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	tmp = 0.0;
	if (n <= -9.2e-149)
		tmp = t_0;
	elseif (n <= -4.9e-209)
		tmp = 0.0;
	elseif (n <= -2.5e-233)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 2.4e-83)
		tmp = 0.0 / (i / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.2e-149], t$95$0, If[LessEqual[n, -4.9e-209], 0.0, If[LessEqual[n, -2.5e-233], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e-83], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.1999999999999999e-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. Step-by-step derivation

      1. associate-*r/ 20.4%

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

      2. sub-neg 20.4%

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

      3. distribute-rgt-in 20.4%

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

      4. metadata-eval 20.4%

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

      5. metadata-eval 20.4%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in n around inf 38.3%

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

   6. Taylor expanded in i around 0 53.3%

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

      1. *-commutative 53.3%

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

   8. Simplified 53.3%

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

   9. Taylor expanded in n around 0 69.1%

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

   if -9.1999999999999999e-149 < n < -4.90000000000000035e-209

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. sub-neg 73.1%

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

      3. distribute-rgt-in 73.1%

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

      4. metadata-eval 73.1%

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

      5. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in i around 0 73.1%

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

   6. Taylor expanded in i around 0 73.1%

      \[\leadsto \color{blue}{0} \]

   if -4.90000000000000035e-209 < n < -2.50000000000000006e-233

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 69.0%

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

   if -2.50000000000000006e-233 < 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 76.6%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-149}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.9 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -2.5 \cdot 10^{-233}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{if}\;n \leq -8.5 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.12 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -2.2 \cdot 10^{-232}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{-86}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))
   (if (<= n -8.5e-149)
     t_0
     (if (<= n -1.12e-208)
       0.0
       (if (<= n -2.2e-232)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.32e-86) (/ 0.0 (/ i n)) t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -8.5e-149) {
		tmp = t_0;
	} else if (n <= -1.12e-208) {
		tmp = 0.0;
	} else if (n <= -2.2e-232) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.32e-86) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    if (n <= (-8.5d-149)) then
        tmp = t_0
    else if (n <= (-1.12d-208)) then
        tmp = 0.0d0
    else if (n <= (-2.2d-232)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 1.32d-86) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -8.5e-149) {
		tmp = t_0;
	} else if (n <= -1.12e-208) {
		tmp = 0.0;
	} else if (n <= -2.2e-232) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.32e-86) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	tmp = 0
	if n <= -8.5e-149:
		tmp = t_0
	elif n <= -1.12e-208:
		tmp = 0.0
	elif n <= -2.2e-232:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.32e-86:
		tmp = 0.0 / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))))
	tmp = 0.0
	if (n <= -8.5e-149)
		tmp = t_0;
	elseif (n <= -1.12e-208)
		tmp = 0.0;
	elseif (n <= -2.2e-232)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.32e-86)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	tmp = 0.0;
	if (n <= -8.5e-149)
		tmp = t_0;
	elseif (n <= -1.12e-208)
		tmp = 0.0;
	elseif (n <= -2.2e-232)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 1.32e-86)
		tmp = 0.0 / (i / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -8.5e-149], t$95$0, If[LessEqual[n, -1.12e-208], 0.0, If[LessEqual[n, -2.2e-232], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.32e-86], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.5000000000000006e-149 or 1.32e-86 < n

   1. Initial program 20.3%

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

      1. associate-*r/ 20.3%

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

      2. sub-neg 20.3%

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

      3. distribute-rgt-in 20.3%

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

      4. metadata-eval 20.3%

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

      5. metadata-eval 20.3%

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

   3. Simplified 20.3%

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

   5. Taylor expanded in n around inf 38.1%

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

   6. Taylor expanded in i around 0 66.9%

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

   7. Taylor expanded in n around 0 67.4%

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

   if -8.5000000000000006e-149 < n < -1.12000000000000005e-208

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. sub-neg 73.1%

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

      3. distribute-rgt-in 73.1%

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

      4. metadata-eval 73.1%

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

      5. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in i around 0 73.1%

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

   6. Taylor expanded in i around 0 73.1%

      \[\leadsto \color{blue}{0} \]

   if -1.12000000000000005e-208 < n < -2.20000000000000002e-232

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 69.0%

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

   if -2.20000000000000002e-232 < n < 1.32e-86

   1. Initial program 54.3%

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

      1. associate-*r/ 54.3%

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

      2. sub-neg 54.3%

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

      3. distribute-rgt-in 54.3%

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

      4. metadata-eval 54.3%

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

      5. metadata-eval 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in i around 0 76.1%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-149}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -1.12 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq -2.2 \cdot 10^{-232}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{-86}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -7.8 \cdot 10^{-224}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-160}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 21000000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= n -5e+59)
     (* (* i n) (/ 100.0 i))
     (if (<= n -7.8e-224)
       t_0
       (if (<= n 3.9e-160)
         (/ 0.0 (/ i n))
         (if (<= n 21000000000000.0) t_0 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -5e+59) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -7.8e-224) {
		tmp = t_0;
	} else if (n <= 3.9e-160) {
		tmp = 0.0 / (i / n);
	} else if (n <= 21000000000000.0) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (i / (i / n))
    if (n <= (-5d+59)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= (-7.8d-224)) then
        tmp = t_0
    else if (n <= 3.9d-160) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 21000000000000.0d0) then
        tmp = t_0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -5e+59) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -7.8e-224) {
		tmp = t_0;
	} else if (n <= 3.9e-160) {
		tmp = 0.0 / (i / n);
	} else if (n <= 21000000000000.0) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -5e+59:
		tmp = (i * n) * (100.0 / i)
	elif n <= -7.8e-224:
		tmp = t_0
	elif n <= 3.9e-160:
		tmp = 0.0 / (i / n)
	elif n <= 21000000000000.0:
		tmp = t_0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -5e+59)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= -7.8e-224)
		tmp = t_0;
	elseif (n <= 3.9e-160)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 21000000000000.0)
		tmp = t_0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -5e+59)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= -7.8e-224)
		tmp = t_0;
	elseif (n <= 3.9e-160)
		tmp = 0.0 / (i / n);
	elseif (n <= 21000000000000.0)
		tmp = t_0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5e+59], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -7.8e-224], t$95$0, If[LessEqual[n, 3.9e-160], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 21000000000000.0], t$95$0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.9999999999999997e59

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. expm1-define 68.7%

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

   5. Simplified 68.7%

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

      1. clear-num 68.6%

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

      2. un-div-inv 68.7%

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

      3. associate-/l/ 93.3%

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

   7. Applied egg-rr 93.3%

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

      1. /-rgt-identity 93.3%

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

      2. associate-/r/ 93.2%

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

      3. associate-/r/ 93.2%

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

      4. associate-/r/ 93.4%

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

      5. /-rgt-identity 93.4%

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

      6. *-commutative 93.4%

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

   9. Simplified 93.4%

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

   10. Taylor expanded in i around 0 58.5%

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

       1. *-commutative 58.5%

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

   12. Simplified 58.5%

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

   if -4.9999999999999997e59 < n < -7.7999999999999996e-224 or 3.89999999999999989e-160 < n < 2.1e13

   1. Initial program 26.6%

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

   3. Taylor expanded in i around 0 65.3%

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

   if -7.7999999999999996e-224 < n < 3.89999999999999989e-160

   1. Initial program 62.0%

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

      1. associate-*r/ 62.0%

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

      2. sub-neg 62.0%

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

      3. distribute-rgt-in 62.0%

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

      4. metadata-eval 62.0%

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

      5. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in i around 0 80.6%

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

   if 2.1e13 < n

   1. Initial program 21.9%

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

      1. associate-*r/ 21.9%

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

      2. sub-neg 21.9%

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

      3. distribute-rgt-in 21.9%

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

      4. metadata-eval 21.9%

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

      5. metadata-eval 21.9%

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

   3. Simplified 21.9%

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

   5. Taylor expanded in n around inf 47.4%

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

   6. Taylor expanded in i around 0 77.8%

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

   7. Taylor expanded in i around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. associate-*r* 73.7%

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

      3. distribute-rgt-out 73.7%

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

      4. *-commutative 73.7%

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

   9. Simplified 73.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -7.8 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-160}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 21000000000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -3.1 \cdot 10^{+59}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-161}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 21000000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= n -3.1e+59)
     (* (* i n) (/ 100.0 i))
     (if (<= n -9.5e-224)
       t_0
       (if (<= n 2.6e-161)
         0.0
         (if (<= n 21000000000000.0) t_0 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -3.1e+59) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -9.5e-224) {
		tmp = t_0;
	} else if (n <= 2.6e-161) {
		tmp = 0.0;
	} else if (n <= 21000000000000.0) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (i / (i / n))
    if (n <= (-3.1d+59)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= (-9.5d-224)) then
        tmp = t_0
    else if (n <= 2.6d-161) then
        tmp = 0.0d0
    else if (n <= 21000000000000.0d0) then
        tmp = t_0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -3.1e+59) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -9.5e-224) {
		tmp = t_0;
	} else if (n <= 2.6e-161) {
		tmp = 0.0;
	} else if (n <= 21000000000000.0) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -3.1e+59:
		tmp = (i * n) * (100.0 / i)
	elif n <= -9.5e-224:
		tmp = t_0
	elif n <= 2.6e-161:
		tmp = 0.0
	elif n <= 21000000000000.0:
		tmp = t_0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -3.1e+59)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= -9.5e-224)
		tmp = t_0;
	elseif (n <= 2.6e-161)
		tmp = 0.0;
	elseif (n <= 21000000000000.0)
		tmp = t_0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -3.1e+59)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= -9.5e-224)
		tmp = t_0;
	elseif (n <= 2.6e-161)
		tmp = 0.0;
	elseif (n <= 21000000000000.0)
		tmp = t_0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.1e+59], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -9.5e-224], t$95$0, If[LessEqual[n, 2.6e-161], 0.0, If[LessEqual[n, 21000000000000.0], t$95$0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.10000000000000015e59

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. expm1-define 68.7%

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

   5. Simplified 68.7%

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

      1. clear-num 68.6%

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

      2. un-div-inv 68.7%

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

      3. associate-/l/ 93.3%

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

   7. Applied egg-rr 93.3%

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

      1. /-rgt-identity 93.3%

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

      2. associate-/r/ 93.2%

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

      3. associate-/r/ 93.2%

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

      4. associate-/r/ 93.4%

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

      5. /-rgt-identity 93.4%

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

      6. *-commutative 93.4%

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

   9. Simplified 93.4%

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

   10. Taylor expanded in i around 0 58.5%

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

       1. *-commutative 58.5%

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

   12. Simplified 58.5%

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

   if -3.10000000000000015e59 < n < -9.5000000000000003e-224 or 2.59999999999999995e-161 < n < 2.1e13

   1. Initial program 26.6%

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

   3. Taylor expanded in i around 0 65.3%

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

   if -9.5000000000000003e-224 < n < 2.59999999999999995e-161

   1. Initial program 62.0%

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

      1. associate-*r/ 62.0%

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

      2. sub-neg 62.0%

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

      3. distribute-rgt-in 62.0%

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

      4. metadata-eval 62.0%

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

      5. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in i around 0 80.6%

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

   6. Taylor expanded in i around 0 80.6%

      \[\leadsto \color{blue}{0} \]

   if 2.1e13 < n

   1. Initial program 21.9%

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

      1. associate-*r/ 21.9%

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

      2. sub-neg 21.9%

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

      3. distribute-rgt-in 21.9%

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

      4. metadata-eval 21.9%

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

      5. metadata-eval 21.9%

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

   3. Simplified 21.9%

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

   5. Taylor expanded in n around inf 47.4%

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

   6. Taylor expanded in i around 0 77.8%

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

   7. Taylor expanded in i around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. associate-*r* 73.7%

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

      3. distribute-rgt-out 73.7%

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

      4. *-commutative 73.7%

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

   9. Simplified 73.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{+59}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-161}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 21000000000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-244}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-237}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -6.2e+77)
     t_0
     (if (<= n -2e-244)
       t_1
       (if (<= n 3.4e-237) 0.0 (if (<= n 3.2e-77) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -6.2e+77) {
		tmp = t_0;
	} else if (n <= -2e-244) {
		tmp = t_1;
	} else if (n <= 3.4e-237) {
		tmp = 0.0;
	} else if (n <= 3.2e-77) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    t_1 = 100.0d0 * (i / (i / n))
    if (n <= (-6.2d+77)) then
        tmp = t_0
    else if (n <= (-2d-244)) then
        tmp = t_1
    else if (n <= 3.4d-237) then
        tmp = 0.0d0
    else if (n <= 3.2d-77) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -6.2e+77) {
		tmp = t_0;
	} else if (n <= -2e-244) {
		tmp = t_1;
	} else if (n <= 3.4e-237) {
		tmp = 0.0;
	} else if (n <= 3.2e-77) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -6.2e+77:
		tmp = t_0
	elif n <= -2e-244:
		tmp = t_1
	elif n <= 3.4e-237:
		tmp = 0.0
	elif n <= 3.2e-77:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -6.2e+77)
		tmp = t_0;
	elseif (n <= -2e-244)
		tmp = t_1;
	elseif (n <= 3.4e-237)
		tmp = 0.0;
	elseif (n <= 3.2e-77)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	t_1 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -6.2e+77)
		tmp = t_0;
	elseif (n <= -2e-244)
		tmp = t_1;
	elseif (n <= 3.4e-237)
		tmp = 0.0;
	elseif (n <= 3.2e-77)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e+77], t$95$0, If[LessEqual[n, -2e-244], t$95$1, If[LessEqual[n, 3.4e-237], 0.0, If[LessEqual[n, 3.2e-77], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.19999999999999997e77 or 3.2e-77 < n

   1. Initial program 18.5%

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

      1. associate-*r/ 18.5%

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

      2. sub-neg 18.5%

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

      3. distribute-rgt-in 18.5%

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

      4. metadata-eval 18.5%

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

      5. metadata-eval 18.5%

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

   3. Simplified 18.5%

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

   5. Taylor expanded in n around inf 41.6%

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

   6. Taylor expanded in i around 0 70.6%

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

   7. Taylor expanded in i around 0 68.0%

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

      1. +-commutative 68.0%

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

      2. associate-*r* 68.0%

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

      3. distribute-rgt-out 68.0%

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

      4. *-commutative 68.0%

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

   9. Simplified 68.0%

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

   if -6.19999999999999997e77 < n < -1.9999999999999999e-244 or 3.4000000000000002e-237 < n < 3.2e-77

   1. Initial program 29.9%

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

   3. Taylor expanded in i around 0 61.4%

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

   if -1.9999999999999999e-244 < n < 3.4000000000000002e-237

   1. Initial program 82.2%

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

      1. associate-*r/ 82.2%

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

      2. sub-neg 82.2%

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

      3. distribute-rgt-in 82.2%

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

      4. metadata-eval 82.2%

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

      5. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in i around 0 93.3%

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

   6. Taylor expanded in i around 0 93.3%

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

Alternative 16: 56.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{+77}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{-86}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.2e+77)
   (* n 100.0)
   (if (<= n -5.2e-247)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 1.32e-86) 0.0 (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.2e+77) {
		tmp = n * 100.0;
	} else if (n <= -5.2e-247) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.32e-86) {
		tmp = 0.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-6.2d+77)) then
        tmp = n * 100.0d0
    else if (n <= (-5.2d-247)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 1.32d-86) then
        tmp = 0.0d0
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6.2e+77) {
		tmp = n * 100.0;
	} else if (n <= -5.2e-247) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.32e-86) {
		tmp = 0.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6.2e+77:
		tmp = n * 100.0
	elif n <= -5.2e-247:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.32e-86:
		tmp = 0.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.2e+77)
		tmp = Float64(n * 100.0);
	elseif (n <= -5.2e-247)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.32e-86)
		tmp = 0.0;
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -6.2e+77)
		tmp = n * 100.0;
	elseif (n <= -5.2e-247)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 1.32e-86)
		tmp = 0.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.2e+77], N[(n * 100.0), $MachinePrecision], If[LessEqual[n, -5.2e-247], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.32e-86], 0.0, N[(n * 100.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.19999999999999997e77 or 1.32e-86 < n

   1. Initial program 18.4%

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

   3. Taylor expanded in i around 0 55.3%

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

      1. *-commutative 55.3%

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

   5. Simplified 55.3%

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

   if -6.19999999999999997e77 < n < -5.2e-247

   1. Initial program 33.8%

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

   3. Taylor expanded in i around 0 57.3%

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

   if -5.2e-247 < n < 1.32e-86

   1. Initial program 53.4%

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

      1. associate-*r/ 53.4%

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

      2. sub-neg 53.4%

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

      3. distribute-rgt-in 53.4%

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

      4. metadata-eval 53.4%

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

      5. metadata-eval 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in i around 0 76.5%

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

   6. Taylor expanded in i around 0 76.5%

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

Alternative 17: 55.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-134} \lor \neg \left(n \leq 2.4 \cdot 10^{-83}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.45e-134) (not (<= n 2.4e-83))) (* n 100.0) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.45e-134) || !(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 <= (-1.45d-134)) .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 <= -1.45e-134) || !(n <= 2.4e-83)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.45e-134) 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 <= -1.45e-134) || !(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 <= -1.45e-134) || ~((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, -1.45e-134], 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 < -1.44999999999999997e-134 or 2.4000000000000001e-83 < n

   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 i around 0 55.6%

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

      1. *-commutative 55.6%

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

   5. Simplified 55.6%

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

   if -1.44999999999999997e-134 < n < 2.4000000000000001e-83

   1. Initial program 51.2%

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

      1. associate-*r/ 51.2%

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

      2. sub-neg 51.2%

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

      3. distribute-rgt-in 51.2%

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

      4. metadata-eval 51.2%

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

      5. metadata-eval 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in i around 0 70.0%

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

   6. Taylor expanded in i around 0 70.0%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-134} \lor \neg \left(n \leq 2.4 \cdot 10^{-83}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 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))))