Compound Interest

Percentage Accurate: 28.5% → 97.4%

Time: 16.9s

Alternatives: 16

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.5%

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

      1. associate-/r/ 28.0%

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

      2. add-exp-log 28.0%

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

      3. expm1-define 28.0%

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

      4. log-pow 41.1%

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

      5. log1p-define 97.3%

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

   4. Applied egg-rr 97.3%

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

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

   1. Initial program 97.6%

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

      1. associate-*r/ 97.7%

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

      2. sub-neg 97.7%

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

      3. distribute-rgt-in 98.1%

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

      4. metadata-eval 98.1%

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

      5. metadata-eval 98.1%

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

   3. Simplified 98.1%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

      1. metadata-eval 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. sub-neg 0.0%

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

      5. associate-*r/ 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-/r/ 1.9%

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

      8. associate-*l* 1.9%

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

      9. add-exp-log 1.9%

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

      10. expm1-define 1.9%

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

      11. log-pow 1.9%

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

      12. log1p-define 1.9%

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

   6. Applied egg-rr 1.9%

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

      1. *-commutative 1.9%

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

      2. clear-num 1.9%

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

      3. un-div-inv 1.9%

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

   8. Applied egg-rr 1.9%

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

   9. Taylor expanded in i around 0 99.9%

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

       1. sub-neg 99.9%

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

       2. associate-*r/ 99.9%

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

       3. metadata-eval 99.9%

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

       4. metadata-eval 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 97.9%

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

Alternative 2: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.00185:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{+38}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{1}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -0.00185)
   (/ (* n (* 100.0 (expm1 i))) i)
   (if (<= n 1.7e+38)
     (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
     (* (* n 100.0) (* (expm1 i) (/ 1.0 i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -0.00185) {
		tmp = (n * (100.0 * expm1(i))) / i;
	} else if (n <= 1.7e+38) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * 100.0) * (expm1(i) * (1.0 / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -0.00185) {
		tmp = (n * (100.0 * Math.expm1(i))) / i;
	} else if (n <= 1.7e+38) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * 100.0) * (Math.expm1(i) * (1.0 / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -0.00185:
		tmp = (n * (100.0 * math.expm1(i))) / i
	elif n <= 1.7e+38:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = (n * 100.0) * (math.expm1(i) * (1.0 / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -0.00185)
		tmp = Float64(Float64(n * Float64(100.0 * expm1(i))) / i);
	elseif (n <= 1.7e+38)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) * Float64(1.0 / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -0.00185], N[(N[(n * N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.7e+38], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.0018500000000000001

   1. Initial program 28.0%

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

      1. associate-/r/ 28.4%

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

      2. associate-*r* 28.5%

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

      3. *-commutative 28.5%

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

      4. associate-*r/ 28.5%

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

      5. sub-neg 28.5%

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

      6. distribute-lft-in 28.5%

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

      7. metadata-eval 28.5%

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

      8. metadata-eval 28.5%

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

      9. metadata-eval 28.5%

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

      10. fma-define 28.5%

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

      11. metadata-eval 28.5%

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

   3. Simplified 28.5%

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

   5. Taylor expanded in n around inf 39.8%

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

      1. sub-neg 39.8%

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

      2. metadata-eval 39.8%

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

      3. metadata-eval 39.8%

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

      4. distribute-lft-in 39.8%

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

      5. metadata-eval 39.8%

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

      6. sub-neg 39.8%

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

      7. expm1-define 92.3%

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

   7. Simplified 92.3%

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

   if -0.0018500000000000001 < n < 1.69999999999999998e38

   1. Initial program 30.5%

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

      1. associate-*r/ 30.5%

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

      2. sub-neg 30.5%

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

      3. distribute-rgt-in 30.6%

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

      4. metadata-eval 30.6%

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

      5. metadata-eval 30.6%

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

   3. Simplified 30.6%

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

      1. metadata-eval 30.6%

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

      2. metadata-eval 30.6%

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

      3. distribute-rgt-in 30.5%

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

      4. sub-neg 30.5%

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

      5. associate-*r/ 30.5%

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

      6. *-commutative 30.5%

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

      7. associate-/r/ 30.0%

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

      8. associate-*l* 30.0%

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

      9. add-exp-log 30.0%

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

      10. expm1-define 30.0%

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

      11. log-pow 53.9%

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

      12. log1p-define 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. *-commutative 86.7%

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

      2. clear-num 86.6%

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

      3. un-div-inv 86.7%

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

   8. Applied egg-rr 86.7%

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

   9. Taylor expanded in i around 0 76.0%

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

       1. sub-neg 76.0%

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

       2. associate-*r/ 76.0%

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

       3. metadata-eval 76.0%

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

       4. metadata-eval 76.0%

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

   11. Simplified 76.0%

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

   if 1.69999999999999998e38 < n

   1. Initial program 24.2%

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

      1. associate-/r/ 24.7%

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

      2. associate-*r* 24.8%

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

      3. *-commutative 24.8%

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

      4. associate-*r/ 24.8%

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

      5. sub-neg 24.8%

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

      6. distribute-lft-in 24.8%

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

      7. metadata-eval 24.8%

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

      8. metadata-eval 24.8%

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

      9. metadata-eval 24.8%

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

      10. fma-define 24.8%

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

      11. metadata-eval 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in n around inf 41.3%

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

      1. sub-neg 41.3%

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

      2. metadata-eval 41.3%

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

      3. metadata-eval 41.3%

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

      4. distribute-lft-in 41.3%

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

      5. metadata-eval 41.3%

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

      6. sub-neg 41.3%

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

      7. expm1-define 96.6%

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

   7. Simplified 96.6%

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

      1. div-inv 96.5%

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

      2. associate-*r* 96.7%

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

      3. associate-*l* 96.6%

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

   9. Applied egg-rr 96.6%

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

Alternative 3: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.00192 \lor \neg \left(n \leq 0.012\right):\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -0.00192) (not (<= n 0.012)))
   (/ (* n (* 100.0 (expm1 i))) i)
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -0.00192) || !(n <= 0.012)) {
		tmp = (n * (100.0 * expm1(i))) / i;
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -0.00192) || !(n <= 0.012)) {
		tmp = (n * (100.0 * Math.expm1(i))) / i;
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -0.00192) or not (n <= 0.012):
		tmp = (n * (100.0 * math.expm1(i))) / i
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -0.00192) || !(n <= 0.012))
		tmp = Float64(Float64(n * Float64(100.0 * expm1(i))) / i);
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -0.00192], N[Not[LessEqual[n, 0.012]], $MachinePrecision]], N[(N[(n * N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -0.00192000000000000005 or 0.012 < n

   1. Initial program 25.8%

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

      1. associate-/r/ 26.3%

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

      2. associate-*r* 26.3%

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

      3. *-commutative 26.3%

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

      4. associate-*r/ 26.3%

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

      5. sub-neg 26.3%

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

      6. distribute-lft-in 26.3%

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

      7. metadata-eval 26.3%

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

      8. metadata-eval 26.3%

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

      9. metadata-eval 26.3%

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

      10. fma-define 26.3%

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

      11. metadata-eval 26.3%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in n around inf 38.7%

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

      1. sub-neg 38.7%

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

      2. metadata-eval 38.7%

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

      3. metadata-eval 38.7%

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

      4. distribute-lft-in 38.7%

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

      5. metadata-eval 38.7%

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

      6. sub-neg 38.7%

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

      7. expm1-define 93.9%

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

   7. Simplified 93.9%

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

   if -0.00192000000000000005 < n < 0.012

   1. Initial program 31.4%

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

      1. associate-*r/ 31.4%

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

      2. sub-neg 31.4%

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

      3. distribute-rgt-in 31.5%

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

      4. metadata-eval 31.5%

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

      5. metadata-eval 31.5%

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

   3. Simplified 31.5%

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

      1. metadata-eval 31.5%

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

      2. metadata-eval 31.5%

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

      3. distribute-rgt-in 31.4%

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

      4. sub-neg 31.4%

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

      5. associate-*r/ 31.4%

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

      6. *-commutative 31.4%

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

      7. associate-/r/ 30.8%

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

      8. associate-*l* 30.8%

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

      9. add-exp-log 30.8%

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

      10. expm1-define 30.8%

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

      11. log-pow 57.3%

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

      12. log1p-define 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. *-commutative 86.8%

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

      2. clear-num 86.7%

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

      3. un-div-inv 86.7%

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

   8. Applied egg-rr 86.7%

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

   9. Taylor expanded in i around 0 75.3%

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

       1. sub-neg 75.3%

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

       2. associate-*r/ 75.3%

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

       3. metadata-eval 75.3%

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

       4. metadata-eval 75.3%

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

   11. Simplified 75.3%

       \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.00192 \lor \neg \left(n \leq 0.012\right):\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-6} \lor \neg \left(i \leq 2.85 \cdot 10^{-24}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.2e-6) (not (<= i 2.85e-24)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.2e-6) || !(i <= 2.85e-24)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2.2e-6) || !(i <= 2.85e-24)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.2e-6) or not (i <= 2.85e-24):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.2e-6) || !(i <= 2.85e-24))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.2e-6], N[Not[LessEqual[i, 2.85e-24]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2.2000000000000001e-6 or 2.85000000000000001e-24 < i

   1. Initial program 53.4%

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

   3. Taylor expanded in n around inf 63.4%

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

      1. expm1-define 65.4%

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

   5. Simplified 65.4%

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

   if -2.2000000000000001e-6 < i < 2.85000000000000001e-24

   1. Initial program 7.5%

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

      1. associate-*r/ 7.5%

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

      2. sub-neg 7.5%

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

      3. distribute-rgt-in 7.5%

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

      4. metadata-eval 7.5%

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

      5. metadata-eval 7.5%

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

   3. Simplified 7.5%

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

      1. metadata-eval 7.5%

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

      2. metadata-eval 7.5%

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

      3. distribute-rgt-in 7.5%

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

      4. sub-neg 7.5%

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

      5. associate-*r/ 7.5%

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

      6. *-commutative 7.5%

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

      7. associate-/r/ 8.0%

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

      8. associate-*l* 8.0%

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

      9. add-exp-log 8.0%

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

      10. expm1-define 8.0%

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

      11. log-pow 20.2%

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

      12. log1p-define 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. *-commutative 74.2%

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

      2. clear-num 74.2%

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

      3. un-div-inv 74.2%

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

   8. Applied egg-rr 74.2%

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

   9. Taylor expanded in i around 0 89.3%

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

       1. sub-neg 89.3%

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

       2. associate-*r/ 89.3%

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

       3. metadata-eval 89.3%

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

       4. metadata-eval 89.3%

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

   11. Simplified 89.3%

       \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-6} \lor \neg \left(i \leq 2.85 \cdot 10^{-24}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -5.9 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-209}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= n -5.9e+56)
     (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
     (if (<= n -3.6e-160)
       t_0
       (if (<= n 3.9e-209)
         (/ 0.0 (/ i n))
         (if (<= n 0.012) t_0 (/ (* n (* i (+ 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 <= -5.9e+56) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= -3.6e-160) {
		tmp = t_0;
	} else if (n <= 3.9e-209) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.012) {
		tmp = t_0;
	} else {
		tmp = (n * (i * (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 <= (-5.9d+56)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= (-3.6d-160)) then
        tmp = t_0
    else if (n <= 3.9d-209) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 0.012d0) then
        tmp = t_0
    else
        tmp = (n * (i * (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 <= -5.9e+56) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= -3.6e-160) {
		tmp = t_0;
	} else if (n <= 3.9e-209) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.012) {
		tmp = t_0;
	} else {
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -5.9e+56:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= -3.6e-160:
		tmp = t_0
	elif n <= 3.9e-209:
		tmp = 0.0 / (i / n)
	elif n <= 0.012:
		tmp = t_0
	else:
		tmp = (n * (i * (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 <= -5.9e+56)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= -3.6e-160)
		tmp = t_0;
	elseif (n <= 3.9e-209)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 0.012)
		tmp = t_0;
	else
		tmp = Float64(Float64(n * Float64(i * 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 <= -5.9e+56)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= -3.6e-160)
		tmp = t_0;
	elseif (n <= 3.9e-209)
		tmp = 0.0 / (i / n);
	elseif (n <= 0.012)
		tmp = t_0;
	else
		tmp = (n * (i * (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, -5.9e+56], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -3.6e-160], t$95$0, If[LessEqual[n, 3.9e-209], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.012], t$95$0, N[(N[(n * N[(i * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.9000000000000001e56

   1. Initial program 23.3%

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

      1. associate-*r/ 23.3%

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

      2. sub-neg 23.3%

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

      3. distribute-rgt-in 23.3%

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

      4. metadata-eval 23.3%

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

      5. metadata-eval 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in n around inf 43.2%

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

   6. Taylor expanded in i around 0 68.4%

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

   7. Taylor expanded in n around 0 68.4%

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

   if -5.9000000000000001e56 < n < -3.5999999999999997e-160 or 3.9e-209 < n < 0.012

   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 i around 0 62.0%

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

   if -3.5999999999999997e-160 < n < 3.9e-209

   1. Initial program 63.4%

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

      1. associate-*r/ 63.4%

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

      2. sub-neg 63.4%

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

      3. distribute-rgt-in 63.4%

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

      4. metadata-eval 63.4%

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

      5. metadata-eval 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in i around 0 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in i around 0 82.3%

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

   if 0.012 < n

   1. Initial program 23.5%

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

      1. associate-/r/ 24.0%

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

      2. associate-*r* 24.0%

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

      3. *-commutative 24.0%

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

      4. associate-*r/ 24.0%

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

      5. sub-neg 24.0%

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

      6. distribute-lft-in 24.0%

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

      7. metadata-eval 24.0%

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

      8. metadata-eval 24.0%

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

      9. metadata-eval 24.0%

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

      10. fma-define 24.0%

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

      11. metadata-eval 24.0%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in n around inf 37.5%

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

      1. sub-neg 37.5%

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

      2. metadata-eval 37.5%

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

      3. metadata-eval 37.5%

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

      4. distribute-lft-in 37.5%

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

      5. metadata-eval 37.5%

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

      6. sub-neg 37.5%

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

      7. expm1-define 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in i around 0 80.8%

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

      1. *-commutative 80.8%

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

   10. Simplified 80.8%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-209}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ t_1 := n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{if}\;n \leq -5 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-205}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n))))
        (t_1 (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))
   (if (<= n -5e+56)
     t_1
     (if (<= n -3.6e-160)
       t_0
       (if (<= n 1.1e-205) (/ 0.0 (/ i n)) (if (<= n 0.012) t_0 t_1))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double t_1 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -5e+56) {
		tmp = t_1;
	} else if (n <= -3.6e-160) {
		tmp = t_0;
	} else if (n <= 1.1e-205) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.012) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = 100.0d0 * (i / (i / n))
    t_1 = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    if (n <= (-5d+56)) then
        tmp = t_1
    else if (n <= (-3.6d-160)) then
        tmp = t_0
    else if (n <= 1.1d-205) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 0.012d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double t_1 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -5e+56) {
		tmp = t_1;
	} else if (n <= -3.6e-160) {
		tmp = t_0;
	} else if (n <= 1.1e-205) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.012) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	t_1 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	tmp = 0
	if n <= -5e+56:
		tmp = t_1
	elif n <= -3.6e-160:
		tmp = t_0
	elif n <= 1.1e-205:
		tmp = 0.0 / (i / n)
	elif n <= 0.012:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	t_1 = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))))
	tmp = 0.0
	if (n <= -5e+56)
		tmp = t_1;
	elseif (n <= -3.6e-160)
		tmp = t_0;
	elseif (n <= 1.1e-205)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 0.012)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i / (i / n));
	t_1 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	tmp = 0.0;
	if (n <= -5e+56)
		tmp = t_1;
	elseif (n <= -3.6e-160)
		tmp = t_0;
	elseif (n <= 1.1e-205)
		tmp = 0.0 / (i / n);
	elseif (n <= 0.012)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5e+56], t$95$1, If[LessEqual[n, -3.6e-160], t$95$0, If[LessEqual[n, 1.1e-205], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.012], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.00000000000000024e56 or 0.012 < n

   1. Initial program 23.4%

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

      1. associate-*r/ 23.4%

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

      2. sub-neg 23.4%

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

      3. distribute-rgt-in 23.4%

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

      4. metadata-eval 23.4%

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

      5. metadata-eval 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in n around inf 39.8%

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

   6. Taylor expanded in i around 0 74.2%

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

   7. Taylor expanded in n around 0 74.2%

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

   if -5.00000000000000024e56 < n < -3.5999999999999997e-160 or 1.10000000000000005e-205 < n < 0.012

   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 i around 0 62.0%

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

   if -3.5999999999999997e-160 < n < 1.10000000000000005e-205

   1. Initial program 63.4%

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

      1. associate-*r/ 63.4%

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

      2. sub-neg 63.4%

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

      3. distribute-rgt-in 63.4%

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

      4. metadata-eval 63.4%

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

      5. metadata-eval 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in i around 0 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in i around 0 82.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-205}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(i \cdot 100\right)\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\frac{t\_0}{i}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* i 100.0))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -2e+58)
     (/ t_0 i)
     (if (<= n -3.5e-160)
       t_1
       (if (<= n 5.5e-212)
         (/ 0.0 (/ i n))
         (if (<= n 2.9e+38) t_1 (/ 1.0 (/ i t_0))))))))

C

double code(double i, double n) {
	double t_0 = n * (i * 100.0);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -2e+58) {
		tmp = t_0 / i;
	} else if (n <= -3.5e-160) {
		tmp = t_1;
	} else if (n <= 5.5e-212) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.9e+38) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (i / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = n * (i * 100.0d0)
    t_1 = 100.0d0 * (i / (i / n))
    if (n <= (-2d+58)) then
        tmp = t_0 / i
    else if (n <= (-3.5d-160)) then
        tmp = t_1
    else if (n <= 5.5d-212) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 2.9d+38) then
        tmp = t_1
    else
        tmp = 1.0d0 / (i / t_0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (i * 100.0);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -2e+58) {
		tmp = t_0 / i;
	} else if (n <= -3.5e-160) {
		tmp = t_1;
	} else if (n <= 5.5e-212) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.9e+38) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (i / t_0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (i * 100.0)
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -2e+58:
		tmp = t_0 / i
	elif n <= -3.5e-160:
		tmp = t_1
	elif n <= 5.5e-212:
		tmp = 0.0 / (i / n)
	elif n <= 2.9e+38:
		tmp = t_1
	else:
		tmp = 1.0 / (i / t_0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(i * 100.0))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -2e+58)
		tmp = Float64(t_0 / i);
	elseif (n <= -3.5e-160)
		tmp = t_1;
	elseif (n <= 5.5e-212)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.9e+38)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(i / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (i * 100.0);
	t_1 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -2e+58)
		tmp = t_0 / i;
	elseif (n <= -3.5e-160)
		tmp = t_1;
	elseif (n <= 5.5e-212)
		tmp = 0.0 / (i / n);
	elseif (n <= 2.9e+38)
		tmp = t_1;
	else
		tmp = 1.0 / (i / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * 100.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2e+58], N[(t$95$0 / i), $MachinePrecision], If[LessEqual[n, -3.5e-160], t$95$1, If[LessEqual[n, 5.5e-212], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.9e+38], t$95$1, N[(1.0 / N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.99999999999999989e58

   1. Initial program 23.6%

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

      1. associate-/r/ 24.2%

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

      2. associate-*r* 24.3%

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

      3. *-commutative 24.3%

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

      4. associate-*r/ 24.3%

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

      5. sub-neg 24.3%

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

      6. distribute-lft-in 24.3%

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

      7. metadata-eval 24.3%

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

      8. metadata-eval 24.3%

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

      9. metadata-eval 24.3%

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

      10. fma-define 24.3%

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

      11. metadata-eval 24.3%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in n around inf 44.2%

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

      1. sub-neg 44.2%

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

      2. metadata-eval 44.2%

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

      3. metadata-eval 44.2%

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

      4. distribute-lft-in 44.2%

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

      5. metadata-eval 44.2%

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

      6. sub-neg 44.2%

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

      7. expm1-define 94.5%

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

   7. Simplified 94.5%

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

   8. Taylor expanded in i around 0 67.4%

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

   if -1.99999999999999989e58 < n < -3.5000000000000003e-160 or 5.49999999999999995e-212 < n < 2.90000000000000007e38

   1. Initial program 20.0%

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

   3. Taylor expanded in i around 0 63.9%

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

   if -3.5000000000000003e-160 < n < 5.49999999999999995e-212

   1. Initial program 63.4%

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

      1. associate-*r/ 63.4%

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

      2. sub-neg 63.4%

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

      3. distribute-rgt-in 63.4%

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

      4. metadata-eval 63.4%

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

      5. metadata-eval 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in i around 0 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in i around 0 82.3%

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

   if 2.90000000000000007e38 < n

   1. Initial program 24.2%

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

      1. associate-*r/ 24.2%

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

      2. sub-neg 24.2%

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

      3. distribute-rgt-in 24.2%

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

      4. metadata-eval 24.2%

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

      5. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in i around 0 3.1%

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

      1. *-commutative 3.1%

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

   7. Simplified 3.1%

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

      1. clear-num 3.1%

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

      2. inv-pow 3.1%

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

      3. +-commutative 3.1%

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

      4. associate-+l+ 33.8%

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

      5. metadata-eval 33.8%

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

   9. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{{\left(\frac{\frac{i}{n}}{i \cdot 100 + 0}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 33.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{i \cdot 100 + 0}}} \]

       2. associate-/l/ 75.0%

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

       3. +-rgt-identity 75.0%

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

   11. Simplified 75.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{n \cdot \left(i \cdot 100\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+249}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e+249)
   (/ (* 100.0 (* i n)) i)
   (if (<= n 0.012)
     (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
     (/
      (*
       n
       (*
        i
        (+
         100.0
         (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))
      i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+249) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 0.012) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.7d+249)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= 0.012d0) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-0.5d0))))
    else
        tmp = (n * (i * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0)))))))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+249) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 0.012) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.7e+249:
		tmp = (100.0 * (i * n)) / i
	elif n <= 0.012:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e+249)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= 0.012)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	else
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.7e+249)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= 0.012)
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	else
		tmp = (n * (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e+249], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 0.012], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6999999999999997e249

   1. Initial program 11.9%

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

      1. associate-/r/ 12.5%

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

      2. associate-*r* 12.6%

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

      3. *-commutative 12.6%

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

      4. associate-*r/ 12.6%

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

      5. sub-neg 12.6%

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

      6. distribute-lft-in 12.6%

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

      7. metadata-eval 12.6%

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

      8. metadata-eval 12.6%

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

      9. metadata-eval 12.6%

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

      10. fma-define 12.6%

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

      11. metadata-eval 12.6%

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

   3. Simplified 12.6%

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

   5. Taylor expanded in n around inf 56.3%

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

      1. sub-neg 56.3%

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

      2. metadata-eval 56.3%

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

      3. metadata-eval 56.3%

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

      4. distribute-lft-in 56.3%

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

      5. metadata-eval 56.3%

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

      6. sub-neg 56.3%

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

      7. expm1-define 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in i around 0 89.3%

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

      1. *-commutative 89.3%

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

   10. Simplified 89.3%

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

   if -3.6999999999999997e249 < n < 0.012

   1. Initial program 31.9%

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

      1. associate-*r/ 31.9%

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

      2. sub-neg 31.9%

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

      3. distribute-rgt-in 32.0%

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

      4. metadata-eval 32.0%

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

      5. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

      1. metadata-eval 32.0%

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

      2. metadata-eval 32.0%

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

      3. distribute-rgt-in 31.9%

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

      4. sub-neg 31.9%

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

      5. associate-*r/ 31.9%

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

      6. *-commutative 31.9%

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

      7. associate-/r/ 31.7%

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

      8. associate-*l* 31.7%

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

      9. add-exp-log 31.7%

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

      10. expm1-define 31.7%

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

      11. log-pow 45.6%

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

      12. log1p-define 81.4%

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

   6. Applied egg-rr 81.4%

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

      1. *-commutative 81.4%

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

      2. clear-num 81.3%

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

      3. un-div-inv 81.4%

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

   8. Applied egg-rr 81.4%

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

   9. Taylor expanded in i around 0 71.9%

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

       1. sub-neg 71.9%

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

       2. associate-*r/ 71.9%

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

       3. metadata-eval 71.9%

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

       4. metadata-eval 71.9%

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

   11. Simplified 71.9%

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

   if 0.012 < n

   1. Initial program 23.5%

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

      1. associate-/r/ 24.0%

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

      2. associate-*r* 24.0%

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

      3. *-commutative 24.0%

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

      4. associate-*r/ 24.0%

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

      5. sub-neg 24.0%

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

      6. distribute-lft-in 24.0%

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

      7. metadata-eval 24.0%

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

      8. metadata-eval 24.0%

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

      9. metadata-eval 24.0%

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

      10. fma-define 24.0%

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

      11. metadata-eval 24.0%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in n around inf 37.5%

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

      1. sub-neg 37.5%

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

      2. metadata-eval 37.5%

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

      3. metadata-eval 37.5%

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

      4. distribute-lft-in 37.5%

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

      5. metadata-eval 37.5%

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

      6. sub-neg 37.5%

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

      7. expm1-define 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in i around 0 84.7%

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

      1. *-commutative 84.7%

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

   10. Simplified 84.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+249}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot \left(i \cdot 100\right)}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1.55 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-207}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* n (* i 100.0)) i)) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -1.55e+59)
     t_0
     (if (<= n -3.5e-160)
       t_1
       (if (<= n 1.35e-207) (/ 0.0 (/ i n)) (if (<= n 3.1e-22) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = (n * (i * 100.0)) / i;
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.55e+59) {
		tmp = t_0;
	} else if (n <= -3.5e-160) {
		tmp = t_1;
	} else if (n <= 1.35e-207) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.1e-22) {
		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 * (i * 100.0d0)) / i
    t_1 = 100.0d0 * (i / (i / n))
    if (n <= (-1.55d+59)) then
        tmp = t_0
    else if (n <= (-3.5d-160)) then
        tmp = t_1
    else if (n <= 1.35d-207) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 3.1d-22) 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 * (i * 100.0)) / i;
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.55e+59) {
		tmp = t_0;
	} else if (n <= -3.5e-160) {
		tmp = t_1;
	} else if (n <= 1.35e-207) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.1e-22) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * (i * 100.0)) / i
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -1.55e+59:
		tmp = t_0
	elif n <= -3.5e-160:
		tmp = t_1
	elif n <= 1.35e-207:
		tmp = 0.0 / (i / n)
	elif n <= 3.1e-22:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * Float64(i * 100.0)) / i)
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -1.55e+59)
		tmp = t_0;
	elseif (n <= -3.5e-160)
		tmp = t_1;
	elseif (n <= 1.35e-207)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 3.1e-22)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n * (i * 100.0)) / i;
	t_1 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -1.55e+59)
		tmp = t_0;
	elseif (n <= -3.5e-160)
		tmp = t_1;
	elseif (n <= 1.35e-207)
		tmp = 0.0 / (i / n);
	elseif (n <= 3.1e-22)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * N[(i * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.55e+59], t$95$0, If[LessEqual[n, -3.5e-160], t$95$1, If[LessEqual[n, 1.35e-207], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.1e-22], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.55000000000000007e59 or 3.10000000000000013e-22 < n

   1. Initial program 23.7%

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

      1. associate-/r/ 24.2%

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

      2. associate-*r* 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r/ 24.2%

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

      5. sub-neg 24.2%

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

      6. distribute-lft-in 24.3%

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

      7. metadata-eval 24.3%

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

      8. metadata-eval 24.3%

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

      9. metadata-eval 24.3%

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

      10. fma-define 24.2%

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

      11. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in n around inf 39.9%

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

      1. sub-neg 39.9%

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

      2. metadata-eval 39.9%

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

      3. metadata-eval 39.9%

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

      4. distribute-lft-in 39.9%

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

      5. metadata-eval 39.9%

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

      6. sub-neg 39.9%

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

      7. expm1-define 93.7%

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

   7. Simplified 93.7%

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

   8. Taylor expanded in i around 0 71.2%

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

   if -1.55000000000000007e59 < n < -3.5000000000000003e-160 or 1.35e-207 < n < 3.10000000000000013e-22

   1. Initial program 20.0%

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

   3. Taylor expanded in i around 0 63.6%

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

   if -3.5000000000000003e-160 < n < 1.35e-207

   1. Initial program 63.4%

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

      1. associate-*r/ 63.4%

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

      2. sub-neg 63.4%

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

      3. distribute-rgt-in 63.4%

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

      4. metadata-eval 63.4%

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

      5. metadata-eval 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in i around 0 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in i around 0 82.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+59}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-160}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-207}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-22}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+249}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e+249)
   (/ (* 100.0 (* i n)) i)
   (if (<= n 0.012)
     (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
     (/ (* n (* i (+ 100.0 (* i 50.0)))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+249) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 0.012) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * (i * (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) :: tmp
    if (n <= (-3.6d+249)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= 0.012d0) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-0.5d0))))
    else
        tmp = (n * (i * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+249) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 0.012) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.6e+249:
		tmp = (100.0 * (i * n)) / i
	elif n <= 0.012:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e+249)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= 0.012)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	else
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.6e+249)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= 0.012)
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	else
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e+249], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 0.012], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(i * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.5999999999999997e249

   1. Initial program 11.9%

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

      1. associate-/r/ 12.5%

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

      2. associate-*r* 12.6%

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

      3. *-commutative 12.6%

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

      4. associate-*r/ 12.6%

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

      5. sub-neg 12.6%

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

      6. distribute-lft-in 12.6%

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

      7. metadata-eval 12.6%

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

      8. metadata-eval 12.6%

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

      9. metadata-eval 12.6%

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

      10. fma-define 12.6%

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

      11. metadata-eval 12.6%

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

   3. Simplified 12.6%

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

   5. Taylor expanded in n around inf 56.3%

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

      1. sub-neg 56.3%

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

      2. metadata-eval 56.3%

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

      3. metadata-eval 56.3%

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

      4. distribute-lft-in 56.3%

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

      5. metadata-eval 56.3%

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

      6. sub-neg 56.3%

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

      7. expm1-define 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in i around 0 89.3%

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

      1. *-commutative 89.3%

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

   10. Simplified 89.3%

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

   if -3.5999999999999997e249 < n < 0.012

   1. Initial program 31.9%

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

      1. associate-*r/ 31.9%

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

      2. sub-neg 31.9%

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

      3. distribute-rgt-in 32.0%

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

      4. metadata-eval 32.0%

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

      5. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

      1. metadata-eval 32.0%

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

      2. metadata-eval 32.0%

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

      3. distribute-rgt-in 31.9%

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

      4. sub-neg 31.9%

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

      5. associate-*r/ 31.9%

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

      6. *-commutative 31.9%

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

      7. associate-/r/ 31.7%

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

      8. associate-*l* 31.7%

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

      9. add-exp-log 31.7%

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

      10. expm1-define 31.7%

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

      11. log-pow 45.6%

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

      12. log1p-define 81.4%

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

   6. Applied egg-rr 81.4%

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

      1. *-commutative 81.4%

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

      2. clear-num 81.3%

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

      3. un-div-inv 81.4%

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

   8. Applied egg-rr 81.4%

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

   9. Taylor expanded in i around 0 71.9%

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

       1. sub-neg 71.9%

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

       2. associate-*r/ 71.9%

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

       3. metadata-eval 71.9%

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

       4. metadata-eval 71.9%

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

   11. Simplified 71.9%

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

   if 0.012 < n

   1. Initial program 23.5%

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

      1. associate-/r/ 24.0%

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

      2. associate-*r* 24.0%

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

      3. *-commutative 24.0%

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

      4. associate-*r/ 24.0%

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

      5. sub-neg 24.0%

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

      6. distribute-lft-in 24.0%

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

      7. metadata-eval 24.0%

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

      8. metadata-eval 24.0%

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

      9. metadata-eval 24.0%

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

      10. fma-define 24.0%

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

      11. metadata-eval 24.0%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in n around inf 37.5%

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

      1. sub-neg 37.5%

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

      2. metadata-eval 37.5%

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

      3. metadata-eval 37.5%

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

      4. distribute-lft-in 37.5%

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

      5. metadata-eval 37.5%

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

      6. sub-neg 37.5%

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

      7. expm1-define 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in i around 0 80.8%

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

      1. *-commutative 80.8%

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

   10. Simplified 80.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+249}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 0.012:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+58} \lor \neg \left(n \leq 1.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.3e+58) (not (<= n 1.5e-21)))
   (/ (* n (* i 100.0)) i)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.3e+58) || !(n <= 1.5e-21)) {
		tmp = (n * (i * 100.0)) / i;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.3d+58)) .or. (.not. (n <= 1.5d-21))) then
        tmp = (n * (i * 100.0d0)) / i
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.3e+58) || !(n <= 1.5e-21)) {
		tmp = (n * (i * 100.0)) / i;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.3e+58) or not (n <= 1.5e-21):
		tmp = (n * (i * 100.0)) / i
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.3e+58) || !(n <= 1.5e-21))
		tmp = Float64(Float64(n * Float64(i * 100.0)) / i);
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.3e+58) || ~((n <= 1.5e-21)))
		tmp = (n * (i * 100.0)) / i;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.3e+58], N[Not[LessEqual[n, 1.5e-21]], $MachinePrecision]], N[(N[(n * N[(i * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.30000000000000002e58 or 1.49999999999999996e-21 < n

   1. Initial program 23.7%

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

      1. associate-/r/ 24.2%

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

      2. associate-*r* 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r/ 24.2%

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

      5. sub-neg 24.2%

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

      6. distribute-lft-in 24.3%

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

      7. metadata-eval 24.3%

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

      8. metadata-eval 24.3%

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

      9. metadata-eval 24.3%

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

      10. fma-define 24.2%

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

      11. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in n around inf 39.9%

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

      1. sub-neg 39.9%

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

      2. metadata-eval 39.9%

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

      3. metadata-eval 39.9%

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

      4. distribute-lft-in 39.9%

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

      5. metadata-eval 39.9%

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

      6. sub-neg 39.9%

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

      7. expm1-define 93.7%

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

   7. Simplified 93.7%

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

   8. Taylor expanded in i around 0 71.2%

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

   if -2.30000000000000002e58 < n < 1.49999999999999996e-21

   1. Initial program 32.8%

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

   3. Taylor expanded in i around 0 62.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+58} \lor \neg \left(n \leq 1.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+56} \lor \neg \left(n \leq 50000000000\right):\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.9e+56) (not (<= n 50000000000.0)))
   (/ (* 100.0 (* i n)) i)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.9e+56) || !(n <= 50000000000.0)) {
		tmp = (100.0 * (i * n)) / i;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-4.9d+56)) .or. (.not. (n <= 50000000000.0d0))) then
        tmp = (100.0d0 * (i * n)) / i
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.9e+56) || !(n <= 50000000000.0)) {
		tmp = (100.0 * (i * n)) / i;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.9e+56) or not (n <= 50000000000.0):
		tmp = (100.0 * (i * n)) / i
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.9e+56) || !(n <= 50000000000.0))
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -4.9e+56) || ~((n <= 50000000000.0)))
		tmp = (100.0 * (i * n)) / i;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.9e+56], N[Not[LessEqual[n, 50000000000.0]], $MachinePrecision]], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.9000000000000003e56 or 5e10 < n

   1. Initial program 23.9%

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

      1. associate-/r/ 24.4%

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

      2. associate-*r* 24.4%

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

      3. *-commutative 24.4%

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

      4. associate-*r/ 24.4%

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

      5. sub-neg 24.4%

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

      6. distribute-lft-in 24.4%

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

      7. metadata-eval 24.4%

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

      8. metadata-eval 24.4%

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

      9. metadata-eval 24.4%

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

      10. fma-define 24.4%

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

      11. metadata-eval 24.4%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in n around inf 41.3%

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

      1. sub-neg 41.3%

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

      2. metadata-eval 41.3%

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

      3. metadata-eval 41.3%

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

      4. distribute-lft-in 41.3%

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

      5. metadata-eval 41.3%

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

      6. sub-neg 41.3%

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

      7. expm1-define 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in i around 0 71.6%

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

      1. *-commutative 71.6%

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

   10. Simplified 71.6%

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

   if -4.9000000000000003e56 < n < 5e10

   1. Initial program 32.4%

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

   3. Taylor expanded in i around 0 62.3%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+56} \lor \neg \left(n \leq 50000000000\right):\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+61}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2e+61)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 2.0) (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2e+61) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2d+61)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 2.0d0) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2e+61) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2e+61:
		tmp = 100.0 * (i / (i / n))
	elif i <= 2.0:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2e+61)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 2.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2e+61)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 2.0)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2e+61], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.0], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.9999999999999999e61

   1. Initial program 75.3%

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

   3. Taylor expanded in i around 0 36.2%

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

   if -1.9999999999999999e61 < i < 2

   1. Initial program 10.1%

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

   3. Taylor expanded in i around 0 77.1%

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

      1. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if 2 < i

   1. Initial program 40.0%

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

   3. Taylor expanded in i around 0 39.1%

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

      1. associate-*r/ 39.1%

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

      2. metadata-eval 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in n around inf 40.3%

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

      1. *-commutative 40.3%

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

   8. Simplified 40.3%

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

   9. Taylor expanded in i around inf 40.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+61}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (if (<= i 2.0) (* n 100.0) (* (* i n) 50.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 2.0d0) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.0:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.0)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.0], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0 60.3%

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

      1. *-commutative 60.3%

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

   5. Simplified 60.3%

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

   if 2 < i

   1. Initial program 40.0%

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

   3. Taylor expanded in i around 0 39.1%

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

      1. associate-*r/ 39.1%

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

      2. metadata-eval 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in n around inf 40.3%

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

      1. *-commutative 40.3%

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

   8. Simplified 40.3%

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

   9. Taylor expanded in i around inf 40.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.2% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

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

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 28.1%

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

3. Taylor expanded in i around 0 49.9%

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

   1. *-commutative 49.9%

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

5. Simplified 49.9%

   \[\leadsto \color{blue}{n \cdot 100} \]
6. Add Preprocessing

Alternative 16: 2.8% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 28.1%

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

3. Taylor expanded in i around 0 42.1%

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

   1. associate-*r/ 42.1%

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

   2. metadata-eval 42.1%

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

5. Simplified 42.1%

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

6. Taylor expanded in n around 0 2.8%

   \[\leadsto \color{blue}{-50 \cdot i} \]
7. Step-by-step derivation

   1. *-commutative 2.8%

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

8. Simplified 2.8%

   \[\leadsto \color{blue}{i \cdot -50} \]
9. Add Preprocessing

Developer target: 34.3% 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 2024106 
(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))))