Compound Interest

Percentage Accurate: 27.8% → 98.2%

Time: 24.4s

Alternatives: 16

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{t_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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)
       (* 100.0 (- (/ t_0 (/ i n)) (/ n i)))
       (/ 1.0 (+ (* (/ i n) -0.005) (* 0.01 (/ 1.0 n))))))))

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 = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	}
	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 = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	}
	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 = 100.0 * ((t_0 / (i / n)) - (n / i))
	else:
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)))
	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(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 / Float64(i / n)) - Float64(n / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(i / n) * -0.005) + Float64(0.01 * Float64(1.0 / n))));
	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[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(i / n), $MachinePrecision] * -0.005), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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 22.1%

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

      1. *-un-lft-identity 22.1%

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

      2. add-exp-log 22.1%

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

      3. expm1-def 22.1%

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

      4. log-pow 29.4%

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

      5. log1p-udef 99.2%

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

   4. Applied egg-rr 99.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. *-lft-identity 99.2%

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

   6. Simplified 99.2%

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

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

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-exp-log 99.9%

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

      3. expm1-def 99.9%

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

      4. log-pow 76.2%

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

      5. log1p-udef 76.2%

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

   4. Applied egg-rr 76.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. *-lft-identity 76.2%

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

   6. Simplified 76.2%

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

      1. expm1-udef 76.2%

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

      2. div-sub 76.2%

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

      3. *-commutative 76.2%

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

      4. log1p-udef 76.2%

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

      5. exp-to-pow 99.9%

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

      6. +-commutative 99.9%

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

      7. clear-num 99.9%

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

   8. Applied egg-rr 99.9%

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

   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. Add Preprocessing

   3. Taylor expanded in n around inf 0.0%

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

      1. expm1-def 4.5%

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

   5. Simplified 4.5%

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

      1. associate-*r/ 4.5%

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

      2. div-inv 4.5%

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

      3. associate-/r* 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. clear-num 78.1%

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

      2. inv-pow 78.1%

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

      3. div-inv 78.0%

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

      4. clear-num 78.1%

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

      5. associate-/r* 77.7%

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

   9. Applied egg-rr 77.7%

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

       1. unpow-1 77.7%

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

       2. associate-*l/ 77.7%

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

       3. *-lft-identity 77.7%

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

       4. associate-/l/ 78.0%

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

       5. *-commutative 78.0%

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

   11. Simplified 78.0%

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

   12. Taylor expanded in i around 0 99.8%

       \[\leadsto \frac{1}{\color{blue}{-0.005 \cdot \frac{i}{n} + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{t_0}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

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)
       (* 100.0 (- (/ t_0 (/ i n)) (/ n i)))
       (/ 1.0 (+ (* (/ i n) -0.005) (* 0.01 (/ 1.0 n))))))))

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 = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	}
	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 = 100.0 * ((t_0 / (i / n)) - (n / i));
	} else {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	}
	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 = 100.0 * ((t_0 / (i / n)) - (n / i))
	else:
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)))
	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(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(n / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 / Float64(i / n)) - Float64(n / i)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(i / n) * -0.005) + Float64(0.01 * Float64(1.0 / n))));
	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[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(i / n), $MachinePrecision] * -0.005), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $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 22.1%

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

      1. clear-num 22.1%

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

      2. associate-/r/ 22.1%

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

      3. clear-num 21.7%

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

      4. add-exp-log 21.7%

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

      5. expm1-def 21.7%

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

      6. log-pow 29.1%

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

      7. log1p-udef 97.8%

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

   4. Applied egg-rr 97.8%

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

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

   1. Initial program 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-exp-log 99.9%

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

      3. expm1-def 99.9%

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

      4. log-pow 76.2%

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

      5. log1p-udef 76.2%

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

   4. Applied egg-rr 76.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. *-lft-identity 76.2%

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

   6. Simplified 76.2%

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

      1. expm1-udef 76.2%

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

      2. div-sub 76.2%

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

      3. *-commutative 76.2%

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

      4. log1p-udef 76.2%

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

      5. exp-to-pow 99.9%

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

      6. +-commutative 99.9%

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

      7. clear-num 99.9%

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

   8. Applied egg-rr 99.9%

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

   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. Add Preprocessing

   3. Taylor expanded in n around inf 0.0%

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

      1. expm1-def 4.5%

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

   5. Simplified 4.5%

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

      1. associate-*r/ 4.5%

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

      2. div-inv 4.5%

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

      3. associate-/r* 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. clear-num 78.1%

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

      2. inv-pow 78.1%

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

      3. div-inv 78.0%

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

      4. clear-num 78.1%

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

      5. associate-/r* 77.7%

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

   9. Applied egg-rr 77.7%

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

       1. unpow-1 77.7%

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

       2. associate-*l/ 77.7%

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

       3. *-lft-identity 77.7%

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

       4. associate-/l/ 78.0%

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

       5. *-commutative 78.0%

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

   11. Simplified 78.0%

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

   12. Taylor expanded in i around 0 99.8%

       \[\leadsto \frac{1}{\color{blue}{-0.005 \cdot \frac{i}{n} + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-201}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}{n}}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 8.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.1e-201)
   (/ 1.0 (/ (/ i (* 100.0 (expm1 i))) n))
   (if (<= n 9.5e-207)
     (/ 0.0 (/ i n))
     (if (<= n 8.4e-20)
       (/
        1.0
        (/ (+ 0.01 (+ (* i -0.005) (* 0.0008333333333333334 (pow i 2.0)))) n))
       (* n (/ 100.0 (/ i (expm1 i))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.1e-201) {
		tmp = 1.0 / ((i / (100.0 * expm1(i))) / n);
	} else if (n <= 9.5e-207) {
		tmp = 0.0 / (i / n);
	} else if (n <= 8.4e-20) {
		tmp = 1.0 / ((0.01 + ((i * -0.005) + (0.0008333333333333334 * pow(i, 2.0)))) / n);
	} else {
		tmp = n * (100.0 / (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.1e-201) {
		tmp = 1.0 / ((i / (100.0 * Math.expm1(i))) / n);
	} else if (n <= 9.5e-207) {
		tmp = 0.0 / (i / n);
	} else if (n <= 8.4e-20) {
		tmp = 1.0 / ((0.01 + ((i * -0.005) + (0.0008333333333333334 * Math.pow(i, 2.0)))) / n);
	} else {
		tmp = n * (100.0 / (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.1e-201:
		tmp = 1.0 / ((i / (100.0 * math.expm1(i))) / n)
	elif n <= 9.5e-207:
		tmp = 0.0 / (i / n)
	elif n <= 8.4e-20:
		tmp = 1.0 / ((0.01 + ((i * -0.005) + (0.0008333333333333334 * math.pow(i, 2.0)))) / n)
	else:
		tmp = n * (100.0 / (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.1e-201)
		tmp = Float64(1.0 / Float64(Float64(i / Float64(100.0 * expm1(i))) / n));
	elseif (n <= 9.5e-207)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 8.4e-20)
		tmp = Float64(1.0 / Float64(Float64(0.01 + Float64(Float64(i * -0.005) + Float64(0.0008333333333333334 * (i ^ 2.0)))) / n));
	else
		tmp = Float64(n * Float64(100.0 / Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.1e-201], N[(1.0 / N[(N[(i / N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-207], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.4e-20], N[(1.0 / N[(N[(0.01 + N[(N[(i * -0.005), $MachinePrecision] + N[(0.0008333333333333334 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.0999999999999999e-201

   1. Initial program 22.1%

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

   3. Taylor expanded in n around inf 33.2%

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

      1. expm1-def 71.8%

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

   5. Simplified 71.8%

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

      1. associate-*r/ 71.7%

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

      2. div-inv 71.5%

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

      3. associate-/r* 84.3%

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

   7. Applied egg-rr 84.3%

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

      1. clear-num 84.8%

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

      2. inv-pow 84.8%

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

      3. div-inv 84.7%

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

      4. clear-num 84.8%

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

      5. associate-/r* 84.7%

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

   9. Applied egg-rr 84.7%

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

       1. unpow-1 84.7%

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

       2. associate-*l/ 84.9%

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

       3. *-lft-identity 84.9%

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

       4. associate-/l/ 84.9%

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

       5. *-commutative 84.9%

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

   11. Simplified 84.9%

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

   if -3.0999999999999999e-201 < n < 9.50000000000000007e-207

   1. Initial program 65.4%

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

      1. associate-*r/ 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-lft-in 65.4%

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

      4. metadata-eval 65.4%

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

      5. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in i around 0 85.4%

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

   if 9.50000000000000007e-207 < n < 8.3999999999999996e-20

   1. Initial program 9.6%

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

   3. Taylor expanded in n around inf 3.3%

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

      1. expm1-def 62.6%

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

   5. Simplified 62.6%

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

      1. associate-*r/ 62.2%

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

      2. div-inv 62.1%

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

      3. associate-/r* 62.7%

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

   7. Applied egg-rr 62.7%

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

      1. clear-num 62.6%

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

      2. inv-pow 62.6%

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

      3. div-inv 62.5%

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

      4. clear-num 62.4%

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

      5. associate-/r* 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. unpow-1 62.5%

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

       2. associate-*l/ 62.6%

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

       3. *-lft-identity 62.6%

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

       4. associate-/l/ 62.5%

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

       5. *-commutative 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in i around 0 84.1%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}}{n}} \]

   if 8.3999999999999996e-20 < n

   1. Initial program 21.7%

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

   3. Taylor expanded in n around inf 47.6%

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

      1. expm1-def 80.7%

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

   5. Simplified 80.7%

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

      1. clear-num 80.7%

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

      2. un-div-inv 80.6%

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

   7. Applied egg-rr 80.6%

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

      1. associate-/l* 80.6%

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

      2. associate-/r/ 98.6%

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

      3. associate-/l* 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-201}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}{n}}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 8.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-207}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.25 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -3.6e-199)
     t_0
     (if (<= n 9e-207)
       (/ 0.0 (/ i n))
       (if (<= n 3.25e-31) (/ 1.0 (/ (+ 0.01 (* i -0.005)) n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -3.6e-199) {
		tmp = t_0;
	} else if (n <= 9e-207) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.25e-31) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -3.6e-199) {
		tmp = t_0;
	} else if (n <= 9e-207) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.25e-31) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -3.6e-199:
		tmp = t_0
	elif n <= 9e-207:
		tmp = 0.0 / (i / n)
	elif n <= 3.25e-31:
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -3.6e-199)
		tmp = t_0;
	elseif (n <= 9e-207)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 3.25e-31)
		tmp = Float64(1.0 / Float64(Float64(0.01 + Float64(i * -0.005)) / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.6e-199], t$95$0, If[LessEqual[n, 9e-207], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.25e-31], N[(1.0 / N[(N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6000000000000002e-199 or 3.24999999999999983e-31 < n

   1. Initial program 21.9%

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

      1. associate-*r/ 22.0%

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

      2. sub-neg 22.0%

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

      3. distribute-lft-in 21.9%

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

      4. metadata-eval 21.9%

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

      5. metadata-eval 21.9%

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

   3. Simplified 21.9%

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

      1. metadata-eval 21.9%

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

      2. metadata-eval 21.9%

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

      3. distribute-lft-in 22.0%

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

      4. sub-neg 22.0%

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

      5. associate-*r/ 21.9%

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

      6. associate-/r/ 22.0%

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

      7. associate-*r* 22.0%

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

      8. add-exp-log 22.0%

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

      9. expm1-def 22.0%

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

      10. log-pow 22.7%

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

      11. log1p-udef 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in n around inf 39.1%

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

      1. expm1-def 90.4%

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

   9. Simplified 90.4%

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

   if -3.6000000000000002e-199 < n < 8.99999999999999984e-207

   1. Initial program 65.4%

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

      1. associate-*r/ 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-lft-in 65.4%

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

      4. metadata-eval 65.4%

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

      5. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in i around 0 85.4%

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

   if 8.99999999999999984e-207 < n < 3.24999999999999983e-31

   1. Initial program 9.6%

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

   3. Taylor expanded in n around inf 3.3%

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

      1. expm1-def 62.6%

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

   5. Simplified 62.6%

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

      1. associate-*r/ 62.2%

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

      2. div-inv 62.1%

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

      3. associate-/r* 62.7%

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

   7. Applied egg-rr 62.7%

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

      1. clear-num 62.6%

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

      2. inv-pow 62.6%

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

      3. div-inv 62.5%

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

      4. clear-num 62.4%

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

      5. associate-/r* 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. unpow-1 62.5%

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

       2. associate-*l/ 62.6%

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

       3. *-lft-identity 62.6%

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

       4. associate-/l/ 62.5%

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

       5. *-commutative 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in i around 0 78.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + -0.005 \cdot i}}{n}} \]
   13. Step-by-step derivation

       1. *-commutative 78.5%

          \[\leadsto \frac{1}{\frac{0.01 + \color{blue}{i \cdot -0.005}}{n}} \]

   14. Simplified 78.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + i \cdot -0.005}}{n}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-199}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-207}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.25 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-202}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.25 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.3e-202)
   (* n (* 100.0 (/ (expm1 i) i)))
   (if (<= n 7.2e-205)
     (/ 0.0 (/ i n))
     (if (<= n 3.25e-31)
       (/ 1.0 (/ (+ 0.01 (* i -0.005)) n))
       (* n (/ 100.0 (/ i (expm1 i))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.3e-202) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else if (n <= 7.2e-205) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.25e-31) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = n * (100.0 / (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.3e-202) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else if (n <= 7.2e-205) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.25e-31) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = n * (100.0 / (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.3e-202:
		tmp = n * (100.0 * (math.expm1(i) / i))
	elif n <= 7.2e-205:
		tmp = 0.0 / (i / n)
	elif n <= 3.25e-31:
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n)
	else:
		tmp = n * (100.0 / (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.3e-202)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	elseif (n <= 7.2e-205)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 3.25e-31)
		tmp = Float64(1.0 / Float64(Float64(0.01 + Float64(i * -0.005)) / n));
	else
		tmp = Float64(n * Float64(100.0 / Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.3e-202], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.2e-205], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.25e-31], N[(1.0 / N[(N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.29999999999999989e-202

   1. Initial program 22.1%

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

      1. associate-*r/ 22.1%

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

      2. sub-neg 22.1%

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

      3. distribute-lft-in 22.1%

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

      4. metadata-eval 22.1%

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

      5. metadata-eval 22.1%

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

   3. Simplified 22.1%

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

      1. metadata-eval 22.1%

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

      2. metadata-eval 22.1%

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

      3. distribute-lft-in 22.1%

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

      4. sub-neg 22.1%

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

      5. associate-*r/ 22.1%

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

      6. associate-/r/ 21.8%

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

      7. associate-*r* 21.8%

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

      8. add-exp-log 21.8%

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

      9. expm1-def 21.8%

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

      10. log-pow 24.0%

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

      11. log1p-udef 81.3%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in n around inf 33.0%

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

      1. expm1-def 84.6%

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

   9. Simplified 84.6%

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

   if -3.29999999999999989e-202 < n < 7.1999999999999997e-205

   1. Initial program 65.4%

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

      1. associate-*r/ 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-lft-in 65.4%

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

      4. metadata-eval 65.4%

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

      5. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in i around 0 85.4%

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

   if 7.1999999999999997e-205 < n < 3.24999999999999983e-31

   1. Initial program 9.6%

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

   3. Taylor expanded in n around inf 3.3%

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

      1. expm1-def 62.6%

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

   5. Simplified 62.6%

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

      1. associate-*r/ 62.2%

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

      2. div-inv 62.1%

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

      3. associate-/r* 62.7%

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

   7. Applied egg-rr 62.7%

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

      1. clear-num 62.6%

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

      2. inv-pow 62.6%

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

      3. div-inv 62.5%

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

      4. clear-num 62.4%

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

      5. associate-/r* 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. unpow-1 62.5%

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

       2. associate-*l/ 62.6%

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

       3. *-lft-identity 62.6%

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

       4. associate-/l/ 62.5%

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

       5. *-commutative 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in i around 0 78.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + -0.005 \cdot i}}{n}} \]
   13. Step-by-step derivation

       1. *-commutative 78.5%

          \[\leadsto \frac{1}{\frac{0.01 + \color{blue}{i \cdot -0.005}}{n}} \]

   14. Simplified 78.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + i \cdot -0.005}}{n}} \]

   if 3.24999999999999983e-31 < n

   1. Initial program 21.7%

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

   3. Taylor expanded in n around inf 47.6%

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

      1. expm1-def 80.7%

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

   5. Simplified 80.7%

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

      1. clear-num 80.7%

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

      2. un-div-inv 80.6%

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

   7. Applied egg-rr 80.6%

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

      1. associate-/l* 80.6%

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

      2. associate-/r/ 98.6%

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

      3. associate-/l* 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-202}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.25 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}{n}}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e-206)
   (/ 1.0 (/ (/ i (* 100.0 (expm1 i))) n))
   (if (<= n 7.2e-205)
     (/ 0.0 (/ i n))
     (if (<= n 7.4e-37)
       (/ 1.0 (/ (+ 0.01 (* i -0.005)) n))
       (* n (/ 100.0 (/ i (expm1 i))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e-206) {
		tmp = 1.0 / ((i / (100.0 * expm1(i))) / n);
	} else if (n <= 7.2e-205) {
		tmp = 0.0 / (i / n);
	} else if (n <= 7.4e-37) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = n * (100.0 / (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e-206) {
		tmp = 1.0 / ((i / (100.0 * Math.expm1(i))) / n);
	} else if (n <= 7.2e-205) {
		tmp = 0.0 / (i / n);
	} else if (n <= 7.4e-37) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = n * (100.0 / (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e-206:
		tmp = 1.0 / ((i / (100.0 * math.expm1(i))) / n)
	elif n <= 7.2e-205:
		tmp = 0.0 / (i / n)
	elif n <= 7.4e-37:
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n)
	else:
		tmp = n * (100.0 / (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e-206)
		tmp = Float64(1.0 / Float64(Float64(i / Float64(100.0 * expm1(i))) / n));
	elseif (n <= 7.2e-205)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 7.4e-37)
		tmp = Float64(1.0 / Float64(Float64(0.01 + Float64(i * -0.005)) / n));
	else
		tmp = Float64(n * Float64(100.0 / Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e-206], N[(1.0 / N[(N[(i / N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.2e-205], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.4e-37], N[(1.0 / N[(N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.39999999999999985e-206

   1. Initial program 22.1%

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

   3. Taylor expanded in n around inf 33.2%

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

      1. expm1-def 71.8%

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

   5. Simplified 71.8%

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

      1. associate-*r/ 71.7%

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

      2. div-inv 71.5%

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

      3. associate-/r* 84.3%

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

   7. Applied egg-rr 84.3%

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

      1. clear-num 84.8%

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

      2. inv-pow 84.8%

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

      3. div-inv 84.7%

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

      4. clear-num 84.8%

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

      5. associate-/r* 84.7%

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

   9. Applied egg-rr 84.7%

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

       1. unpow-1 84.7%

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

       2. associate-*l/ 84.9%

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

       3. *-lft-identity 84.9%

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

       4. associate-/l/ 84.9%

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

       5. *-commutative 84.9%

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

   11. Simplified 84.9%

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

   if -3.39999999999999985e-206 < n < 7.1999999999999997e-205

   1. Initial program 65.4%

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

      1. associate-*r/ 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-lft-in 65.4%

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

      4. metadata-eval 65.4%

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

      5. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in i around 0 85.4%

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

   if 7.1999999999999997e-205 < n < 7.4e-37

   1. Initial program 9.6%

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

   3. Taylor expanded in n around inf 3.3%

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

      1. expm1-def 62.6%

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

   5. Simplified 62.6%

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

      1. associate-*r/ 62.2%

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

      2. div-inv 62.1%

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

      3. associate-/r* 62.7%

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

   7. Applied egg-rr 62.7%

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

      1. clear-num 62.6%

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

      2. inv-pow 62.6%

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

      3. div-inv 62.5%

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

      4. clear-num 62.4%

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

      5. associate-/r* 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. unpow-1 62.5%

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

       2. associate-*l/ 62.6%

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

       3. *-lft-identity 62.6%

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

       4. associate-/l/ 62.5%

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

       5. *-commutative 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in i around 0 78.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + -0.005 \cdot i}}{n}} \]
   13. Step-by-step derivation

       1. *-commutative 78.5%

          \[\leadsto \frac{1}{\frac{0.01 + \color{blue}{i \cdot -0.005}}{n}} \]

   14. Simplified 78.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + i \cdot -0.005}}{n}} \]

   if 7.4e-37 < n

   1. Initial program 21.7%

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

   3. Taylor expanded in n around inf 47.6%

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

      1. expm1-def 80.7%

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

   5. Simplified 80.7%

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

      1. clear-num 80.7%

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

      2. un-div-inv 80.6%

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

   7. Applied egg-rr 80.6%

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

      1. associate-/l* 80.6%

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

      2. associate-/r/ 98.6%

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

      3. associate-/l* 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}{n}}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.42 \cdot 10^{-8} \lor \neg \left(i \leq 5.5 \cdot 10^{-34}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.42e-8) (not (<= i 5.5e-34)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (* 100.0 (+ n (* (* i n) (- 0.5 (/ 0.5 n)))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.42e-8) || !(i <= 5.5e-34)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.42e-8) || !(i <= 5.5e-34)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.42e-8) or not (i <= 5.5e-34):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.42e-8) || !(i <= 5.5e-34))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(Float64(i * n) * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.42e-8], N[Not[LessEqual[i, 5.5e-34]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(N[(i * n), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.41999999999999998e-8 or 5.50000000000000014e-34 < i

   1. Initial program 44.0%

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

   3. Taylor expanded in n around inf 67.3%

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

      1. expm1-def 71.4%

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

   5. Simplified 71.4%

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

   if -1.41999999999999998e-8 < i < 5.50000000000000014e-34

   1. Initial program 9.4%

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

   3. Taylor expanded in i around 0 88.0%

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

      1. associate-*r* 88.4%

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

      2. associate-*r/ 88.4%

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

      3. metadata-eval 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.42 \cdot 10^{-8} \lor \neg \left(i \leq 5.5 \cdot 10^{-34}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{-205}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.52 \cdot 10^{-204}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ (+ 0.01 (* i -0.005)) n))))
   (if (<= n -2.4e-205)
     t_0
     (if (<= n 1.52e-204)
       (/ 0.0 (/ i n))
       (if (<= n 3.9e+32) t_0 (* 100.0 (+ n (* n (* i 0.5)))))))))

C

double code(double i, double n) {
	double t_0 = 1.0 / ((0.01 + (i * -0.005)) / n);
	double tmp;
	if (n <= -2.4e-205) {
		tmp = t_0;
	} else if (n <= 1.52e-204) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.9e+32) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (n + (n * (i * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / ((0.01d0 + (i * (-0.005d0))) / n)
    if (n <= (-2.4d-205)) then
        tmp = t_0
    else if (n <= 1.52d-204) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 3.9d+32) then
        tmp = t_0
    else
        tmp = 100.0d0 * (n + (n * (i * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 / ((0.01 + (i * -0.005)) / n);
	double tmp;
	if (n <= -2.4e-205) {
		tmp = t_0;
	} else if (n <= 1.52e-204) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.9e+32) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (n + (n * (i * 0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 1.0 / ((0.01 + (i * -0.005)) / n)
	tmp = 0
	if n <= -2.4e-205:
		tmp = t_0
	elif n <= 1.52e-204:
		tmp = 0.0 / (i / n)
	elif n <= 3.9e+32:
		tmp = t_0
	else:
		tmp = 100.0 * (n + (n * (i * 0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 / Float64(Float64(0.01 + Float64(i * -0.005)) / n))
	tmp = 0.0
	if (n <= -2.4e-205)
		tmp = t_0;
	elseif (n <= 1.52e-204)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 3.9e+32)
		tmp = t_0;
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(i * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 / ((0.01 + (i * -0.005)) / n);
	tmp = 0.0;
	if (n <= -2.4e-205)
		tmp = t_0;
	elseif (n <= 1.52e-204)
		tmp = 0.0 / (i / n);
	elseif (n <= 3.9e+32)
		tmp = t_0;
	else
		tmp = 100.0 * (n + (n * (i * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 / N[(N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.4e-205], t$95$0, If[LessEqual[n, 1.52e-204], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.9e+32], t$95$0, N[(100.0 * N[(n + N[(n * N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.4000000000000002e-205 or 1.51999999999999997e-204 < n < 3.8999999999999999e32

   1. Initial program 18.8%

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

   3. Taylor expanded in n around inf 26.8%

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

      1. expm1-def 72.5%

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

   5. Simplified 72.5%

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

      1. associate-*r/ 72.3%

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

      2. div-inv 72.2%

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

      3. associate-/r* 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. clear-num 82.0%

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

      2. inv-pow 82.0%

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

      3. div-inv 82.0%

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

      4. clear-num 82.0%

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

      5. associate-/r* 82.0%

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

   9. Applied egg-rr 82.0%

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

       1. unpow-1 82.0%

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

       2. associate-*l/ 82.1%

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

       3. *-lft-identity 82.1%

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

       4. associate-/l/ 82.1%

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

       5. *-commutative 82.1%

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

   11. Simplified 82.1%

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

   12. Taylor expanded in i around 0 70.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + -0.005 \cdot i}}{n}} \]
   13. Step-by-step derivation

       1. *-commutative 70.5%

          \[\leadsto \frac{1}{\frac{0.01 + \color{blue}{i \cdot -0.005}}{n}} \]

   14. Simplified 70.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + i \cdot -0.005}}{n}} \]

   if -2.4000000000000002e-205 < n < 1.51999999999999997e-204

   1. Initial program 65.4%

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

      1. associate-*r/ 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-lft-in 65.4%

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

      4. metadata-eval 65.4%

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

      5. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in i around 0 85.4%

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

   if 3.8999999999999999e32 < n

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 55.0%

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

      1. expm1-def 77.7%

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

   5. Simplified 77.7%

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

   6. Taylor expanded in i around 0 72.3%

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

      1. associate-*r* 72.3%

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

   8. Simplified 72.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{elif}\;n \leq 1.52 \cdot 10^{-204}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+32} \lor \neg \left(n \leq 0.1\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9e+32) (not (<= n 0.1)))
   (* 100.0 (/ (* i n) i))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9e+32) || !(n <= 0.1)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-9d+32)) .or. (.not. (n <= 0.1d0))) then
        tmp = 100.0d0 * ((i * n) / i)
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9e+32) || !(n <= 0.1)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9e+32) or not (n <= 0.1):
		tmp = 100.0 * ((i * n) / i)
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9e+32) || !(n <= 0.1))
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -9e+32) || ~((n <= 0.1)))
		tmp = 100.0 * ((i * n) / i);
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9e+32], N[Not[LessEqual[n, 0.1]], $MachinePrecision]], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.0000000000000007e32 or 0.10000000000000001 < n

   1. Initial program 18.4%

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

   3. Taylor expanded in i around 0 5.8%

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

      1. +-commutative 5.8%

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

   5. Simplified 5.8%

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

      1. div-inv 5.7%

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

      2. associate--l+ 32.7%

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

      3. metadata-eval 32.7%

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

      4. +-rgt-identity 32.7%

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

      5. clear-num 32.8%

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

      6. *-commutative 32.8%

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

   7. Applied egg-rr 32.8%

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

      1. associate-*l/ 67.5%

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

   9. Applied egg-rr 67.5%

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

   if -9.0000000000000007e32 < n < 0.10000000000000001

   1. Initial program 33.5%

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

   3. Taylor expanded in i around 0 66.9%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+32} \lor \neg \left(n \leq 0.1\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 7.3 \cdot 10^{-22}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6e+32)
   (* 100.0 (/ (* i n) i))
   (if (<= n 7.3e-22) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6e+32) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 7.3e-22) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-6d+32)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 7.3d-22) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6e+32) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 7.3e-22) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6e+32:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 7.3e-22:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6e+32)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 7.3e-22)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -6e+32)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 7.3e-22)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6e+32], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.3e-22], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6e32

   1. Initial program 13.2%

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

   3. Taylor expanded in i around 0 4.9%

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

      1. +-commutative 4.9%

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

   5. Simplified 4.9%

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

      1. div-inv 4.8%

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

      2. associate--l+ 31.5%

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

      3. metadata-eval 31.5%

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

      4. +-rgt-identity 31.5%

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

      5. clear-num 31.6%

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

      6. *-commutative 31.6%

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

   7. Applied egg-rr 31.6%

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

      1. associate-*l/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   if -6e32 < n < 7.30000000000000028e-22

   1. Initial program 35.0%

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

   3. Taylor expanded in i around 0 65.3%

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

   if 7.30000000000000028e-22 < n

   1. Initial program 21.7%

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

   3. Taylor expanded in n around inf 47.6%

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

      1. expm1-def 80.7%

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

   5. Simplified 80.7%

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

   6. Taylor expanded in i around 0 75.3%

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

      1. associate-*r* 75.3%

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

      2. distribute-rgt-out 75.3%

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

   8. Simplified 75.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 7.3 \cdot 10^{-22}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 3.9e+32)
   (/ 1.0 (/ (+ 0.01 (* i -0.005)) n))
   (* 100.0 (+ n (* n (* i 0.5))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 3.9e+32) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = 100.0 * (n + (n * (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 <= 3.9d+32) then
        tmp = 1.0d0 / ((0.01d0 + (i * (-0.005d0))) / n)
    else
        tmp = 100.0d0 * (n + (n * (i * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 3.9e+32) {
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	} else {
		tmp = 100.0 * (n + (n * (i * 0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 3.9e+32:
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n)
	else:
		tmp = 100.0 * (n + (n * (i * 0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 3.9e+32)
		tmp = Float64(1.0 / Float64(Float64(0.01 + Float64(i * -0.005)) / n));
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(i * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 3.9e+32)
		tmp = 1.0 / ((0.01 + (i * -0.005)) / n);
	else
		tmp = 100.0 * (n + (n * (i * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 3.9e+32], N[(1.0 / N[(N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(n * N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.8999999999999999e32

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf 29.4%

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

      1. expm1-def 65.2%

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

   5. Simplified 65.2%

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

      1. associate-*r/ 65.0%

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

      2. div-inv 64.9%

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

      3. associate-/r* 73.2%

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

   7. Applied egg-rr 73.2%

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

      1. clear-num 73.5%

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

      2. inv-pow 73.5%

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

      3. div-inv 73.5%

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

      4. clear-num 73.5%

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

      5. associate-/r* 73.5%

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

   9. Applied egg-rr 73.5%

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

       1. unpow-1 73.5%

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

       2. associate-*l/ 73.6%

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

       3. *-lft-identity 73.6%

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

       4. associate-/l/ 73.6%

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

       5. *-commutative 73.6%

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

   11. Simplified 73.6%

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

   12. Taylor expanded in i around 0 66.2%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + -0.005 \cdot i}}{n}} \]
   13. Step-by-step derivation

       1. *-commutative 66.2%

          \[\leadsto \frac{1}{\frac{0.01 + \color{blue}{i \cdot -0.005}}{n}} \]

   14. Simplified 66.2%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.01 + i \cdot -0.005}}{n}} \]

   if 3.8999999999999999e32 < n

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 55.0%

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

      1. expm1-def 77.7%

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

   5. Simplified 77.7%

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

   6. Taylor expanded in i around 0 72.3%

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

      1. associate-*r* 72.3%

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

   8. Simplified 72.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\frac{0.01 + i \cdot -0.005}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+144}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2e+144)
   (* 100.0 (* i (/ n i)))
   (if (<= i 2.0) (* n 100.0) (* n (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2e+144) {
		tmp = 100.0 * (i * (n / i));
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (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 (i <= (-2d+144)) then
        tmp = 100.0d0 * (i * (n / i))
    else if (i <= 2.0d0) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2e+144) {
		tmp = 100.0 * (i * (n / i));
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2e+144:
		tmp = 100.0 * (i * (n / i))
	elif i <= 2.0:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2e+144)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	elseif (i <= 2.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2e+144)
		tmp = 100.0 * (i * (n / i));
	elseif (i <= 2.0)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2e+144], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.0], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.00000000000000005e144

   1. Initial program 72.8%

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

   3. Taylor expanded in i around 0 46.1%

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

      1. +-commutative 46.1%

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

   5. Simplified 46.1%

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

      1. div-inv 46.1%

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

      2. associate--l+ 46.1%

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

      3. metadata-eval 46.1%

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

      4. +-rgt-identity 46.1%

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

      5. clear-num 42.3%

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

      6. *-commutative 42.3%

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

   7. Applied egg-rr 42.3%

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

   if -2.00000000000000005e144 < i < 2

   1. Initial program 12.2%

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

   3. Taylor expanded in i around 0 73.5%

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

      1. *-commutative 73.5%

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

   5. Simplified 73.5%

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

   if 2 < i

   1. Initial program 45.2%

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

   3. Taylor expanded in n around inf 64.4%

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

      1. expm1-def 64.4%

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

   5. Simplified 64.4%

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

   6. Taylor expanded in i around 0 40.6%

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

      1. associate-*r* 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in i around inf 40.6%

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

       1. associate-*r* 40.6%

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

       2. *-commutative 40.6%

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

   11. Simplified 40.6%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+144}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+20}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1e+20)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 2.0) (* n 100.0) (* n (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1e+20) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (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 (i <= (-1d+20)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 2.0d0) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1e+20) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1e+20:
		tmp = 100.0 * (i / (i / n))
	elif i <= 2.0:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1e+20)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 2.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1e+20)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 2.0)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1e+20], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.0], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1e20

   1. Initial program 57.2%

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

   3. Taylor expanded in i around 0 28.6%

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

   if -1e20 < i < 2

   1. Initial program 9.3%

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

   3. Taylor expanded in i around 0 81.6%

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

      1. *-commutative 81.6%

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

   5. Simplified 81.6%

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

   if 2 < i

   1. Initial program 45.2%

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

   3. Taylor expanded in n around inf 64.4%

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

      1. expm1-def 64.4%

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

   5. Simplified 64.4%

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

   6. Taylor expanded in i around 0 40.6%

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

      1. associate-*r* 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in i around inf 40.6%

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

       1. associate-*r* 40.6%

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

       2. *-commutative 40.6%

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

   11. Simplified 40.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+20}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = n * (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 (i <= 2.0d0) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 2

   1. Initial program 19.6%

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

   3. Taylor expanded in i around 0 65.0%

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

      1. *-commutative 65.0%

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

   5. Simplified 65.0%

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

   if 2 < i

   1. Initial program 45.2%

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

   3. Taylor expanded in n around inf 64.4%

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

      1. expm1-def 64.4%

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

   5. Simplified 64.4%

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

   6. Taylor expanded in i around 0 40.6%

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

      1. associate-*r* 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in i around inf 40.6%

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

       1. associate-*r* 40.6%

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

       2. *-commutative 40.6%

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

   11. Simplified 40.6%

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

4. Final simplification 59.9%

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

Alternative 15: 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 24.9%

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

   1. associate-*r/ 24.9%

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

   2. sub-neg 24.9%

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

   3. distribute-lft-in 24.9%

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

   4. metadata-eval 24.9%

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

   5. metadata-eval 24.9%

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

3. Simplified 24.9%

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

   1. metadata-eval 24.9%

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

   2. metadata-eval 24.9%

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

   3. distribute-lft-in 24.9%

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

   4. sub-neg 24.9%

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

   5. associate-*r/ 24.9%

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

   6. associate-/r/ 25.0%

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

   7. associate-*r* 25.0%

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

   8. add-exp-log 25.0%

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

   9. expm1-def 25.0%

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

   10. log-pow 28.7%

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

   11. log1p-udef 81.1%

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

6. Applied egg-rr 81.1%

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

7. Taylor expanded in i around 0 59.5%

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

   1. associate-*r/ 59.5%

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

   2. metadata-eval 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in n around 0 2.9%

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

    1. *-commutative 2.9%

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

12. Simplified 2.9%

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

13. Final simplification 2.9%

    \[\leadsto i \cdot -50 \]
14. Add Preprocessing

Alternative 16: 49.8% 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 24.9%

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

3. Taylor expanded in i around 0 52.7%

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

   1. *-commutative 52.7%

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

5. Simplified 52.7%

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

6. Final simplification 52.7%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 33.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(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))))