Compound Interest

Percentage Accurate: 28.1% → 98.2%

Time: 35.9s

Alternatives: 18

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: 28.1% 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: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

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 (expm1 (* n (log1p (/ i n))))) (/ i n))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * Math.expm1((n * Math.log1p((i / n))))) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (100.0 * math.expm1((n * math.log1p((i / n))))) / (i / n)
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 29.5%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

      1. metadata-eval 29.5%

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

      2. sub-neg 29.5%

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

      3. associate-/r/ 29.5%

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

      4. associate-*r/ 29.5%

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

      5. add-exp-log 29.5%

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

      6. expm1-def 29.5%

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

      7. log-pow 39.6%

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

      8. log1p-udef 98.1%

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

   6. Applied egg-rr 98.1%

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

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. fma-def 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t\_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 0.0)
     (* 100.0 (/ n (/ i (expm1 i))))
     (if (<= t_1 INFINITY)
       (* 100.0 (* n (/ t_0 i)))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * (t_0 / i))
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(t_0 / i)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 29.5%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in n around inf 39.8%

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

      1. associate-/l* 39.8%

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

      2. expm1-def 77.2%

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

   7. Simplified 77.2%

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

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

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 (/ i (expm1 i))))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 29.5%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in n around inf 39.8%

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

      1. associate-/l* 39.8%

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

      2. expm1-def 77.2%

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

   7. Simplified 77.2%

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

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. fma-def 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

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 (expm1 (* n (log1p (/ i n))))) (/ n i))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) * (n / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * Math.expm1((n * Math.log1p((i / n))))) * (n / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (100.0 * math.expm1((n * math.log1p((i / n))))) * (n / i)
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) * Float64(n / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 29.5%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

      1. metadata-eval 29.5%

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

      2. sub-neg 29.5%

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

      3. associate-/r/ 29.5%

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

      4. associate-*r/ 29.5%

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

      5. sub-neg 29.5%

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

      6. metadata-eval 29.5%

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

      7. distribute-lft-in 29.5%

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

      8. metadata-eval 29.5%

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

      9. fma-udef 29.5%

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

      10. associate-/r/ 29.5%

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

      11. *-commutative 29.5%

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

      12. clear-num 29.5%

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

      13. un-div-inv 29.5%

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

      14. fma-udef 29.5%

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

      15. metadata-eval 29.5%

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

      16. distribute-lft-in 29.5%

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

   6. Applied egg-rr 96.9%

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

      1. associate-/r/ 96.1%

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

   8. Simplified 96.1%

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

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. fma-def 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i \cdot 0.01}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

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)
     (* (expm1 (* n (log1p (/ i n)))) (/ n (* i 0.01)))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = expm1((n * log1p((i / n)))) * (n / (i * 0.01));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) * (n / (i * 0.01));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.expm1((n * math.log1p((i / n)))) * (n / (i * 0.01))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(n / Float64(i * 0.01)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / N[(i * 0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 29.5%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

      1. metadata-eval 29.5%

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

      2. sub-neg 29.5%

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

      3. associate-/r/ 29.5%

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

      4. associate-*r/ 29.5%

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

      5. sub-neg 29.5%

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

      6. metadata-eval 29.5%

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

      7. distribute-lft-in 29.5%

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

      8. metadata-eval 29.5%

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

      9. fma-udef 29.5%

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

      10. associate-/r/ 29.5%

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

      11. *-commutative 29.5%

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

      12. clear-num 29.5%

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

      13. un-div-inv 29.5%

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

      14. fma-udef 29.5%

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

      15. metadata-eval 29.5%

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

      16. distribute-lft-in 29.5%

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

   6. Applied egg-rr 96.9%

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

      1. associate-/r* 96.8%

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

      2. associate-/r/ 96.1%

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

      3. div-inv 96.1%

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

      4. metadata-eval 96.1%

         \[\leadsto \frac{n}{i \cdot \color{blue}{0.01}} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \]

   8. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\frac{n}{i \cdot 0.01} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)} \]

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. fma-def 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i \cdot 0.01}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

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)
     (/ n (/ i (* 100.0 (expm1 (* n (log1p (/ i n)))))))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n / (i / (100.0 * expm1((n * log1p((i / n))))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n / (i / (100.0 * Math.expm1((n * Math.log1p((i / n))))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = n / (i / (100.0 * math.expm1((n * math.log1p((i / n))))))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n / Float64(i / Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n)))))));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(n / N[(i / N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 29.5%

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

      1. associate-/r/ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

      1. metadata-eval 29.5%

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

      2. sub-neg 29.5%

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

      3. associate-/r/ 29.5%

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

      4. associate-*r/ 29.5%

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

      5. sub-neg 29.5%

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

      6. metadata-eval 29.5%

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

      7. distribute-lft-in 29.5%

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

      8. metadata-eval 29.5%

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

      9. fma-udef 29.5%

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

      10. associate-/r/ 29.5%

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

      11. *-commutative 29.5%

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

      12. clear-num 29.5%

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

      13. un-div-inv 29.5%

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

      14. fma-udef 29.5%

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

      15. metadata-eval 29.5%

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

      16. distribute-lft-in 29.5%

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

   6. Applied egg-rr 96.9%

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

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. fma-def 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -1.2 \cdot 10^{-210}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-223}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) - 0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -1.2e-210)
     t_0
     (if (<= n 4.5e-223)
       (* (/ 0.0 i) (* n 100.0))
       (if (<= n 7e-41)
         (/ (/ (* n 100.0) i) (- (+ (* i 0.08333333333333333) (/ 1.0 i)) 0.5))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -1.2e-210) {
		tmp = t_0;
	} else if (n <= 4.5e-223) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 7e-41) {
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -1.2e-210) {
		tmp = t_0;
	} else if (n <= 4.5e-223) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 7e-41) {
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -1.2e-210:
		tmp = t_0
	elif n <= 4.5e-223:
		tmp = (0.0 / i) * (n * 100.0)
	elif n <= 7e-41:
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -1.2e-210)
		tmp = t_0;
	elseif (n <= 4.5e-223)
		tmp = Float64(Float64(0.0 / i) * Float64(n * 100.0));
	elseif (n <= 7e-41)
		tmp = Float64(Float64(Float64(n * 100.0) / i) / Float64(Float64(Float64(i * 0.08333333333333333) + Float64(1.0 / i)) - 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.2e-210], t$95$0, If[LessEqual[n, 4.5e-223], N[(N[(0.0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-41], N[(N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision] / N[(N[(N[(i * 0.08333333333333333), $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.20000000000000002e-210 or 6.9999999999999999e-41 < n

   1. Initial program 27.1%

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

      1. associate-/r/ 27.4%

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

      2. sub-neg 27.4%

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

      3. metadata-eval 27.4%

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

   3. Simplified 27.4%

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

   5. Taylor expanded in n around inf 39.2%

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

      1. associate-/l* 39.2%

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

      2. expm1-def 87.7%

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

   7. Simplified 87.7%

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

   if -1.20000000000000002e-210 < n < 4.49999999999999968e-223

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/r/ 70.1%

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

      3. associate-*l* 70.1%

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

      4. sub-neg 70.1%

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

      5. metadata-eval 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in i around 0 85.5%

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

   if 4.49999999999999968e-223 < n < 6.9999999999999999e-41

   1. Initial program 18.4%

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

      1. associate-/r/ 18.7%

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

      2. sub-neg 18.7%

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

      3. metadata-eval 18.7%

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

   3. Simplified 18.7%

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

   5. Taylor expanded in n around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. expm1-def 39.7%

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

   7. Simplified 39.7%

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

      1. associate-*r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. div-inv 39.6%

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

      4. associate-/r* 39.4%

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

   9. Applied egg-rr 39.4%

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

   10. Taylor expanded in i around 0 64.7%

       \[\leadsto \frac{\frac{n \cdot 100}{i}}{\color{blue}{\left(0.08333333333333333 \cdot i + \frac{1}{i}\right) - 0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-223}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) - 0.5}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{-209}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) - 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.7e-209)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= n 3.8e-222)
     (* (/ 0.0 i) (* n 100.0))
     (if (<= n 7e-41)
       (/ (/ (* n 100.0) i) (- (+ (* i 0.08333333333333333) (/ 1.0 i)) 0.5))
       (* (* n 100.0) (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.7e-209) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (n <= 3.8e-222) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 7e-41) {
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5);
	} else {
		tmp = (n * 100.0) * (expm1(i) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.7e-209) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (n <= 3.8e-222) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 7e-41) {
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5);
	} else {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.7e-209:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif n <= 3.8e-222:
		tmp = (0.0 / i) * (n * 100.0)
	elif n <= 7e-41:
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5)
	else:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.7e-209)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (n <= 3.8e-222)
		tmp = Float64(Float64(0.0 / i) * Float64(n * 100.0));
	elseif (n <= 7e-41)
		tmp = Float64(Float64(Float64(n * 100.0) / i) / Float64(Float64(Float64(i * 0.08333333333333333) + Float64(1.0 / i)) - 0.5));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.7e-209], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-222], N[(N[(0.0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-41], N[(N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision] / N[(N[(N[(i * 0.08333333333333333), $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.69999999999999994e-209

   1. Initial program 30.1%

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

      1. associate-/r/ 30.4%

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

      2. sub-neg 30.4%

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

      3. metadata-eval 30.4%

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

   3. Simplified 30.4%

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

   5. Taylor expanded in n around inf 38.6%

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

      1. associate-/l* 38.6%

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

      2. expm1-def 82.8%

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

   7. Simplified 82.8%

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

   if -1.69999999999999994e-209 < n < 3.79999999999999997e-222

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/r/ 70.1%

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

      3. associate-*l* 70.1%

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

      4. sub-neg 70.1%

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

      5. metadata-eval 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in i around 0 85.5%

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

   if 3.79999999999999997e-222 < n < 6.9999999999999999e-41

   1. Initial program 18.4%

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

      1. associate-/r/ 18.7%

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

      2. sub-neg 18.7%

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

      3. metadata-eval 18.7%

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

   3. Simplified 18.7%

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

   5. Taylor expanded in n around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. expm1-def 39.7%

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

   7. Simplified 39.7%

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

      1. associate-*r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. div-inv 39.6%

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

      4. associate-/r* 39.4%

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

   9. Applied egg-rr 39.4%

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

   10. Taylor expanded in i around 0 64.7%

       \[\leadsto \frac{\frac{n \cdot 100}{i}}{\color{blue}{\left(0.08333333333333333 \cdot i + \frac{1}{i}\right) - 0.5}} \]

   if 6.9999999999999999e-41 < n

   1. Initial program 22.8%

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

      1. *-commutative 22.8%

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

      2. associate-/r/ 23.1%

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

      3. associate-*l* 23.1%

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

      4. sub-neg 23.1%

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

      5. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in n around inf 39.9%

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

      1. expm1-def 94.7%

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

   7. Simplified 94.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{-209}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) - 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{elif}\;n \leq -7 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-222}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) - 0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.9e+178)
   (* (* n 100.0) (+ 1.0 (* i 0.5)))
   (if (<= n -7e-210)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 2.5e-222)
       (* (/ 0.0 i) (* n 100.0))
       (if (<= n 7e-41)
         (/ (/ (* n 100.0) i) (- (+ (* i 0.08333333333333333) (/ 1.0 i)) 0.5))
         (* n (+ 100.0 (* i 50.0))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.9e+178) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else if (n <= -7e-210) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 2.5e-222) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 7e-41) {
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.9d+178)) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    else if (n <= (-7d-210)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 2.5d-222) then
        tmp = (0.0d0 / i) * (n * 100.0d0)
    else if (n <= 7d-41) then
        tmp = ((n * 100.0d0) / i) / (((i * 0.08333333333333333d0) + (1.0d0 / i)) - 0.5d0)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.9e+178) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else if (n <= -7e-210) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 2.5e-222) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 7e-41) {
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.9e+178:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	elif n <= -7e-210:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 2.5e-222:
		tmp = (0.0 / i) * (n * 100.0)
	elif n <= 7e-41:
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.9e+178)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	elseif (n <= -7e-210)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 2.5e-222)
		tmp = Float64(Float64(0.0 / i) * Float64(n * 100.0));
	elseif (n <= 7e-41)
		tmp = Float64(Float64(Float64(n * 100.0) / i) / Float64(Float64(Float64(i * 0.08333333333333333) + Float64(1.0 / i)) - 0.5));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.9e+178)
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	elseif (n <= -7e-210)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 2.5e-222)
		tmp = (0.0 / i) * (n * 100.0);
	elseif (n <= 7e-41)
		tmp = ((n * 100.0) / i) / (((i * 0.08333333333333333) + (1.0 / i)) - 0.5);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.9e+178], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -7e-210], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-222], N[(N[(0.0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-41], N[(N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision] / N[(N[(N[(i * 0.08333333333333333), $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -4.9000000000000001e178

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

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

      2. associate-/r/ 16.2%

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

      3. associate-*l* 16.2%

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

      4. sub-neg 16.2%

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

      5. metadata-eval 16.2%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in i around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in n around inf 60.7%

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

      1. *-commutative 60.7%

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

   10. Simplified 60.7%

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

   if -4.9000000000000001e178 < n < -7.00000000000000031e-210

   1. Initial program 36.9%

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

      1. associate-/r/ 37.0%

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

      2. sub-neg 37.0%

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

      3. metadata-eval 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in n around inf 29.2%

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

      1. associate-/l* 29.2%

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

      2. expm1-def 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in i around 0 66.6%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 66.6%

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

   10. Simplified 66.6%

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

   if -7.00000000000000031e-210 < n < 2.50000000000000004e-222

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/r/ 70.1%

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

      3. associate-*l* 70.1%

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

      4. sub-neg 70.1%

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

      5. metadata-eval 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in i around 0 85.5%

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

   if 2.50000000000000004e-222 < n < 6.9999999999999999e-41

   1. Initial program 18.4%

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

      1. associate-/r/ 18.7%

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

      2. sub-neg 18.7%

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

      3. metadata-eval 18.7%

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

   3. Simplified 18.7%

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

   5. Taylor expanded in n around inf 3.4%

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

      1. associate-/l* 3.4%

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

      2. expm1-def 39.7%

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

   7. Simplified 39.7%

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

      1. associate-*r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. div-inv 39.6%

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

      4. associate-/r* 39.4%

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

   9. Applied egg-rr 39.4%

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

   10. Taylor expanded in i around 0 64.7%

       \[\leadsto \frac{\frac{n \cdot 100}{i}}{\color{blue}{\left(0.08333333333333333 \cdot i + \frac{1}{i}\right) - 0.5}} \]

   if 6.9999999999999999e-41 < n

   1. Initial program 22.8%

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

      1. associate-/r/ 23.1%

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

      2. sub-neg 23.1%

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

      3. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in n around inf 39.9%

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

      1. associate-/l* 39.9%

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

      2. expm1-def 94.7%

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

   7. Simplified 94.7%

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

      1. associate-*r/ 94.7%

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

      2. *-commutative 94.7%

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

      3. div-inv 94.6%

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

      4. associate-/r* 78.4%

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

   9. Applied egg-rr 78.4%

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

   10. Taylor expanded in i around 0 73.9%

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

       1. associate-*r* 73.9%

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

       2. distribute-rgt-out 73.9%

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

   12. Simplified 73.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{elif}\;n \leq -7 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-222}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{n \cdot 100}{i}}{\left(i \cdot 0.08333333333333333 + \frac{1}{i}\right) - 0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{+178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{elif}\;n \leq -1.32 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-152}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.8e+178)
   (* (* n 100.0) (+ 1.0 (* i 0.5)))
   (if (<= n -1.32e-210)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 1.35e-152)
       (* (/ 0.0 i) (* n 100.0))
       (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.8e+178) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else if (n <= -1.32e-210) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.35e-152) {
		tmp = (0.0 / i) * (n * 100.0);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.8d+178)) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    else if (n <= (-1.32d-210)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1.35d-152) then
        tmp = (0.0d0 / i) * (n * 100.0d0)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.8e+178) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else if (n <= -1.32e-210) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.35e-152) {
		tmp = (0.0 / i) * (n * 100.0);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.8e+178:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	elif n <= -1.32e-210:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.35e-152:
		tmp = (0.0 / i) * (n * 100.0)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.8e+178)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	elseif (n <= -1.32e-210)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.35e-152)
		tmp = Float64(Float64(0.0 / i) * Float64(n * 100.0));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.8e+178)
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	elseif (n <= -1.32e-210)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1.35e-152)
		tmp = (0.0 / i) * (n * 100.0);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.8e+178], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.32e-210], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.35e-152], N[(N[(0.0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.8e178

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

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

      2. associate-/r/ 16.2%

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

      3. associate-*l* 16.2%

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

      4. sub-neg 16.2%

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

      5. metadata-eval 16.2%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in i around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in n around inf 60.7%

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

      1. *-commutative 60.7%

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

   10. Simplified 60.7%

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

   if -4.8e178 < n < -1.3200000000000001e-210

   1. Initial program 36.9%

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

      1. associate-/r/ 37.0%

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

      2. sub-neg 37.0%

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

      3. metadata-eval 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in n around inf 29.2%

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

      1. associate-/l* 29.2%

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

      2. expm1-def 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in i around 0 66.6%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 66.6%

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

   10. Simplified 66.6%

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

   if -1.3200000000000001e-210 < n < 1.34999999999999999e-152

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-/r/ 59.6%

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

      3. associate-*l* 59.6%

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

      4. sub-neg 59.6%

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

      5. metadata-eval 59.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in i around 0 77.1%

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

   if 1.34999999999999999e-152 < n

   1. Initial program 19.7%

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

      1. associate-/r/ 20.0%

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

      2. sub-neg 20.0%

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

      3. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in n around inf 29.7%

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

      1. associate-/l* 29.7%

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

      2. expm1-def 80.2%

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

   7. Simplified 80.2%

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

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. div-inv 80.1%

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

      4. associate-/r* 68.4%

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

   9. Applied egg-rr 68.4%

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

   10. Taylor expanded in i around 0 65.4%

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

       1. associate-*r* 65.4%

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

       2. distribute-rgt-out 65.4%

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

   12. Simplified 65.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{+178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{elif}\;n \leq -1.32 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-152}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{+178} \lor \neg \left(n \leq 10^{-67}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.45e+178) (not (<= n 1e-67)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.45e+178) || !(n <= 1e-67)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.45d+178)) .or. (.not. (n <= 1d-67))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.45e+178) || !(n <= 1e-67)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.45e+178) or not (n <= 1e-67):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.45e+178) || !(n <= 1e-67))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.45e+178) || ~((n <= 1e-67)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.45e+178], N[Not[LessEqual[n, 1e-67]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.45e178 or 9.99999999999999943e-68 < n

   1. Initial program 19.6%

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

      1. associate-/r/ 20.0%

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

      2. sub-neg 20.0%

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

      3. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in n around inf 43.3%

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

      1. associate-/l* 43.3%

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

      2. expm1-def 92.3%

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

   7. Simplified 92.3%

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

      1. associate-*r/ 92.3%

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

      2. *-commutative 92.3%

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

      3. div-inv 92.2%

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

      4. associate-/r* 72.2%

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

   9. Applied egg-rr 72.2%

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

   10. Taylor expanded in i around 0 68.3%

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

       1. associate-*r* 68.3%

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

       2. distribute-rgt-out 68.3%

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

   12. Simplified 68.3%

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

   if -1.45e178 < n < 9.99999999999999943e-68

   1. Initial program 40.5%

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

      1. associate-/r/ 40.7%

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

      2. sub-neg 40.7%

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

      3. metadata-eval 40.7%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in n around inf 29.5%

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

      1. associate-/l* 29.5%

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

      2. expm1-def 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in i around 0 62.6%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 62.6%

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

   10. Simplified 62.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{+178} \lor \neg \left(n \leq 10^{-67}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-69}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.5e+178)
   (* (* n 100.0) (+ 1.0 (* i 0.5)))
   (if (<= n 2.4e-69)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+178) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else if (n <= 2.4e-69) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.5d+178)) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    else if (n <= 2.4d-69) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+178) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else if (n <= 2.4e-69) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.5e+178:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	elif n <= 2.4e-69:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.5e+178)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	elseif (n <= 2.4e-69)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.5e+178)
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	elseif (n <= 2.4e-69)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.5e+178], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e-69], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.50000000000000008e178

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

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

      2. associate-/r/ 16.2%

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

      3. associate-*l* 16.2%

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

      4. sub-neg 16.2%

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

      5. metadata-eval 16.2%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in i around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in n around inf 60.7%

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

      1. *-commutative 60.7%

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

   10. Simplified 60.7%

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

   if -1.50000000000000008e178 < n < 2.4000000000000001e-69

   1. Initial program 40.5%

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

      1. associate-/r/ 40.7%

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

      2. sub-neg 40.7%

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

      3. metadata-eval 40.7%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in n around inf 29.5%

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

      1. associate-/l* 29.5%

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

      2. expm1-def 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in i around 0 62.6%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 62.6%

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

   10. Simplified 62.6%

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

   if 2.4000000000000001e-69 < n

   1. Initial program 21.2%

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

      1. associate-/r/ 21.5%

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

      2. sub-neg 21.5%

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

      3. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

   5. Taylor expanded in n around inf 36.8%

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

      1. associate-/l* 36.8%

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

      2. expm1-def 90.5%

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

   7. Simplified 90.5%

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

      1. associate-*r/ 90.5%

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

      2. *-commutative 90.5%

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

      3. div-inv 90.3%

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

      4. associate-/r* 75.6%

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

   9. Applied egg-rr 75.6%

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

   10. Taylor expanded in i around 0 71.5%

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

       1. associate-*r* 71.5%

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

       2. distribute-rgt-out 71.5%

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

   12. Simplified 71.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-69}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{+56} \lor \neg \left(n \leq 2.5 \cdot 10^{-33}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{n}{i \cdot 0.01}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.7e+56) (not (<= n 2.5e-33)))
   (* n (+ 100.0 (* i 50.0)))
   (* i (/ n (* i 0.01)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.7e+56) || !(n <= 2.5e-33)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = i * (n / (i * 0.01));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-4.7d+56)) .or. (.not. (n <= 2.5d-33))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = i * (n / (i * 0.01d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.7e+56) || !(n <= 2.5e-33)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = i * (n / (i * 0.01));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.7e+56) or not (n <= 2.5e-33):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = i * (n / (i * 0.01))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.7e+56) || !(n <= 2.5e-33))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(i * Float64(n / Float64(i * 0.01)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -4.7e+56) || ~((n <= 2.5e-33)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = i * (n / (i * 0.01));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.7e+56], N[Not[LessEqual[n, 2.5e-33]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(n / N[(i * 0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.7e56 or 2.50000000000000014e-33 < n

   1. Initial program 23.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. sub-neg 24.2%

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

      3. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in n around inf 42.9%

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

      1. associate-/l* 42.9%

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

      2. expm1-def 92.6%

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

   7. Simplified 92.6%

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

      1. associate-*r/ 92.6%

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

      2. *-commutative 92.6%

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

      3. div-inv 92.4%

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

      4. associate-/r* 71.3%

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

   9. Applied egg-rr 71.3%

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

   10. Taylor expanded in i around 0 67.1%

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

       1. associate-*r* 67.1%

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

       2. distribute-rgt-out 67.1%

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

   12. Simplified 67.1%

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

   if -4.7e56 < n < 2.50000000000000014e-33

   1. Initial program 39.3%

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

      1. associate-/r/ 39.4%

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

      2. sub-neg 39.4%

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

      3. metadata-eval 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in n around inf 27.5%

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

      1. associate-/l* 27.5%

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

      2. expm1-def 54.5%

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

   7. Simplified 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. div-inv 54.4%

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

      4. associate-/r* 54.2%

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

   9. Applied egg-rr 54.2%

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

   10. Taylor expanded in i around 0 58.9%

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

       1. associate-/r/ 59.0%

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

       2. clear-num 60.5%

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

       3. clear-num 60.5%

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

       4. clear-num 59.0%

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

       5. associate-/l* 58.9%

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

       6. div-inv 58.9%

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

       7. metadata-eval 58.9%

          \[\leadsto \frac{n}{i \cdot \color{blue}{0.01}} \cdot i \]

   12. Applied egg-rr 58.9%

       \[\leadsto \color{blue}{\frac{n}{i \cdot 0.01} \cdot i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{+56} \lor \neg \left(n \leq 2.5 \cdot 10^{-33}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{n}{i \cdot 0.01}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+50} \lor \neg \left(n \leq 5.1 \cdot 10^{-33}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{i}{\frac{n}{0.01}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.6e+50) (not (<= n 5.1e-33)))
   (* n (+ 100.0 (* i 50.0)))
   (/ i (/ i (/ n 0.01)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.6e+50) || !(n <= 5.1e-33)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = i / (i / (n / 0.01));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-5.6d+50)) .or. (.not. (n <= 5.1d-33))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = i / (i / (n / 0.01d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5.6e+50) || !(n <= 5.1e-33)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = i / (i / (n / 0.01));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5.6e+50) or not (n <= 5.1e-33):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = i / (i / (n / 0.01))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.6e+50) || !(n <= 5.1e-33))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(i / Float64(i / Float64(n / 0.01)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -5.6e+50) || ~((n <= 5.1e-33)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = i / (i / (n / 0.01));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.6e+50], N[Not[LessEqual[n, 5.1e-33]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i / N[(i / N[(n / 0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.5999999999999996e50 or 5.10000000000000008e-33 < n

   1. Initial program 23.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. sub-neg 24.2%

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

      3. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in n around inf 42.9%

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

      1. associate-/l* 42.9%

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

      2. expm1-def 92.6%

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

   7. Simplified 92.6%

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

      1. associate-*r/ 92.6%

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

      2. *-commutative 92.6%

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

      3. div-inv 92.4%

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

      4. associate-/r* 71.3%

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

   9. Applied egg-rr 71.3%

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

   10. Taylor expanded in i around 0 67.1%

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

       1. associate-*r* 67.1%

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

       2. distribute-rgt-out 67.1%

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

   12. Simplified 67.1%

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

   if -5.5999999999999996e50 < n < 5.10000000000000008e-33

   1. Initial program 39.3%

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

      1. associate-/r/ 39.4%

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

      2. sub-neg 39.4%

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

      3. metadata-eval 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in n around inf 27.5%

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

      1. associate-/l* 27.5%

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

      2. expm1-def 54.5%

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

   7. Simplified 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. div-inv 54.4%

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

      4. associate-/r* 54.2%

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

   9. Applied egg-rr 54.2%

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

   10. Taylor expanded in i around 0 58.9%

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

       1. associate-/r/ 59.0%

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

       2. /-rgt-identity 59.0%

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

       3. *-commutative 59.0%

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

       4. associate-/l* 58.9%

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

       5. div-inv 58.9%

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

       6. metadata-eval 58.9%

          \[\leadsto i \cdot \frac{n}{i \cdot \color{blue}{0.01}} \]

   12. Applied egg-rr 58.9%

       \[\leadsto \color{blue}{i \cdot \frac{n}{i \cdot 0.01}} \]
   13. Step-by-step derivation

       1. associate-*r/ 25.3%

          \[\leadsto \color{blue}{\frac{i \cdot n}{i \cdot 0.01}} \]

       2. associate-/l* 60.4%

          \[\leadsto \color{blue}{\frac{i}{\frac{i \cdot 0.01}{n}}} \]

       3. associate-/l* 60.6%

          \[\leadsto \frac{i}{\color{blue}{\frac{i}{\frac{n}{0.01}}}} \]

   14. Simplified 60.6%

       \[\leadsto \color{blue}{\frac{i}{\frac{i}{\frac{n}{0.01}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+50} \lor \neg \left(n \leq 5.1 \cdot 10^{-33}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{i}{\frac{n}{0.01}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{+55} \lor \neg \left(n \leq 3 \cdot 10^{-35}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.12e+55) (not (<= n 3e-35)))
   (* n (+ 100.0 (* i 50.0)))
   (/ (* i 100.0) (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.12e+55) || !(n <= 3e-35)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (i * 100.0) / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.12d+55)) .or. (.not. (n <= 3d-35))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = (i * 100.0d0) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.12e+55) || !(n <= 3e-35)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (i * 100.0) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.12e+55) or not (n <= 3e-35):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = (i * 100.0) / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.12e+55) || !(n <= 3e-35))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(Float64(i * 100.0) / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.12e+55) || ~((n <= 3e-35)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = (i * 100.0) / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.12e+55], N[Not[LessEqual[n, 3e-35]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.12000000000000006e55 or 2.99999999999999989e-35 < n

   1. Initial program 23.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. sub-neg 24.2%

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

      3. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in n around inf 42.9%

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

      1. associate-/l* 42.9%

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

      2. expm1-def 92.6%

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

   7. Simplified 92.6%

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

      1. associate-*r/ 92.6%

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

      2. *-commutative 92.6%

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

      3. div-inv 92.4%

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

      4. associate-/r* 71.3%

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

   9. Applied egg-rr 71.3%

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

   10. Taylor expanded in i around 0 67.1%

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

       1. associate-*r* 67.1%

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

       2. distribute-rgt-out 67.1%

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

   12. Simplified 67.1%

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

   if -1.12000000000000006e55 < n < 2.99999999999999989e-35

   1. Initial program 39.3%

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

      1. associate-/r/ 39.4%

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

      2. sub-neg 39.4%

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

      3. metadata-eval 39.4%

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

   3. Simplified 39.4%

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

      1. metadata-eval 39.4%

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

      2. sub-neg 39.4%

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

      3. associate-/r/ 39.3%

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

      4. associate-*r/ 39.3%

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

      5. add-exp-log 39.3%

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

      6. expm1-def 39.3%

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

      7. log-pow 56.7%

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

      8. log1p-udef 88.8%

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

   6. Applied egg-rr 88.8%

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

   7. Taylor expanded in i around 0 61.2%

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

      1. *-commutative 61.2%

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

   9. Simplified 61.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{+55} \lor \neg \left(n \leq 3 \cdot 10^{-35}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.5% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ n \cdot \left(100 + i \cdot 50\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n (+ 100.0 (* i 50.0))))

C

double code(double i, double n) {
	return n * (100.0 + (i * 50.0));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * (100.0d0 + (i * 50.0d0))
end function

Java

public static double code(double i, double n) {
	return n * (100.0 + (i * 50.0));
}

Python

def code(i, n):
	return n * (100.0 + (i * 50.0))

Julia

function code(i, n)
	return Float64(n * Float64(100.0 + Float64(i * 50.0)))
end

MATLAB

function tmp = code(i, n)
	tmp = n * (100.0 + (i * 50.0));
end

Wolfram

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

Derivation

1. Initial program 31.0%

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

   1. associate-/r/ 31.3%

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

   2. sub-neg 31.3%

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

   3. metadata-eval 31.3%

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

3. Simplified 31.3%

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

5. Taylor expanded in n around inf 35.7%

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

   1. associate-/l* 35.7%

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

   2. expm1-def 74.9%

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

7. Simplified 74.9%

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

   1. associate-*r/ 74.9%

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

   2. *-commutative 74.9%

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

   3. div-inv 74.7%

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

   4. associate-/r* 63.4%

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

9. Applied egg-rr 63.4%

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

10. Taylor expanded in i around 0 53.0%

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

    1. associate-*r* 53.0%

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

    2. distribute-rgt-out 53.0%

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

12. Simplified 53.0%

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

13. Final simplification 53.0%

    \[\leadsto n \cdot \left(100 + i \cdot 50\right) \]
14. Add Preprocessing

Alternative 17: 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 31.0%

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

   1. *-commutative 31.0%

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

   2. associate-/r/ 31.3%

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

   3. associate-*l* 31.3%

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

   4. sub-neg 31.3%

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

   5. metadata-eval 31.3%

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

3. Simplified 31.3%

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

5. Taylor expanded in i around 0 52.6%

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

   1. associate-*r/ 52.6%

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

   2. metadata-eval 52.6%

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

7. Simplified 52.6%

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

8. Taylor expanded in n around 0 2.9%

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

   1. *-commutative 2.9%

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

10. Simplified 2.9%

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

11. Final simplification 2.9%

    \[\leadsto i \cdot -50 \]
12. Add Preprocessing

Alternative 18: 49.5% 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 31.0%

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

   1. associate-/r/ 31.3%

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

   2. sub-neg 31.3%

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

   3. metadata-eval 31.3%

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

3. Simplified 31.3%

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

5. Taylor expanded in i around 0 46.7%

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

   1. *-commutative 46.7%

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

7. Simplified 46.7%

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

8. Final simplification 46.7%

   \[\leadsto n \cdot 100 \]
9. Add Preprocessing

Developer target: 34.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024101 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 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))))